7.05 Numerical Methods for S. J. Zhong, University of Colorado, Boulder, CO, USA D. A. Yuen, University of Minnesota, Minneapolis, MN, USA L. N. Moresi, Monash University, Clayton, VIC, Australia

ª 2007 Elsevier B.V. All rights reserved.

7.05.1 Introduction 227 7.05.2 Governing Equations and Initial and Boundary Conditions 228 7.05.3 Finite-Difference, Finite-Volume, and Spectral Methods 229 7.05.3.1 Finite Difference 229 7.05.3.1.1 FD implementation of the governing equations 229 7.05.3.1.2 Approximations of spatial derivatives and solution approaches 230 7.05.3.2 FV Method 232 7.05.3.3 Spectral Methods 233 7.05.4 An FE Method 234 7.05.4.1 Stokes Flow: A Weak Formulation, Its FE Implementation, and Solution 234 7.05.4.1.1 A weak formulation 234 7.05.4.1.2 An FE implementation 236 7.05.4.1.3 The Uzawa algorithm for the matrix equation 238 7.05.4.1.4 Multigrid solution strategies 239 7.05.4.2 Stokes Flow: A Penalty Formulation 241 7.05.4.3 The SUPG Formulation for the Energy Equation 242 7.05.5 Incorporating More Realistic Physics 244 7.05.5.1 Thermochemical Convection 244 7.05.5.1.1 Governing equations 244 7.05.5.1.2 Solution approaches 244 7.05.5.2 Solid-State Phase Transition 246 7.05.5.3 Non-Newtonian Rheology 247 7.05.6 Concluding Remarks and Future Prospects 247 References 249

7.05.1 Introduction the field of mantle convection into its own niche in geophysical fluid dynamics (e.g., Yuen et al., 2000). The governing equations for mantle convection are In this chapter, we will present several commonly derived from conservation laws of mass, momentum, used numerical methods in studies of mantle convec- and energy. The nonlinear nature of mantle rheology tion with the primary aim of reaching out to students with its strong temperature and stress dependence and new researchers in the field. First, we will present and nonlinear coupling between flow velocity and the governing equations and the boundary and initial temperature in the energy equation require that conditions for a given problem in mantle convection, numerical methods be used to solve these governing and discuss the general efficient strategy to solve this equations. Understanding the dynamical effects of problem numerically (Section 7.05.2). We will then phase transitions (e.g., -to-spinel phase transi- briefly discuss finite-difference (FD), finite-volume tion) and multicomponent flow also demands (FV), and spectral methods in Section 7.05.3. Since numerical methods. Numerical modeling of mantle finite elements (FEs) have attained very high popu- convection has a rich history since the late 1960s (e.g., larity in the user community, we will discuss FEs Torrance and Turcotte, 1971; McKenzie et al., 1974). in greater details as the most basic numerical Great progress in computer architecture along with tool (Section 7.05.4). For simplicity and clarity, improved numerical techniques has helped advance we will focus our discussion on homogeneous,

227 228 Numerical Methods for Mantle Convection incompressible fluids with the Boussinesq approxi- energy equation. Boundary conditions are in general mation. However, we will also describe methods for a combination of prescribed stress and velocity for more complicated and realistic mantle situations by the momentum equation, and of prescribed heat flux including non-Newtonian rheology, solid-state phase and temperature for the energy equation. The initial transitions, and thermochemical (i.e., multicompo- and boundary conditions can be expressed as nent) convection (Section 7.05.5). Finally, in Section 7.05.6, we will discuss some new developments in Tðri ; t ¼ 0Þ¼Tinitðri Þ½5 computational sciences, such as software develop- ui ¼ gi on g i ;ij nj ¼ hi on hi ½6 ment and visualization, which may impact our ; future studies of mantle convection modeling. T ¼ Tbd on Tbd ðT;i Þn ¼ q on q ½7

where g i and hi are the boundaries where ith components of velocity and forces are specified to 7.05.2 Governing Equations and be g and h , respectively, n is the normal vector of Initial and Boundary Conditions i i j boundary hi , and Tbd and q are the boundaries where temperature and heat flux are prescribed to be The simplest mathematical formulation for mantle Tbd and q, respectively. convection assumes incompressibility and the Often free-slip (i.e., zero tangential stresses and Boussinesq approximation (e.g., McKenzie et al., zero normal velocities) and isothermal conditions 1974). Under this formulation, the nondimensional are applied to the surface and bottom boundaries in conservation equations of the mass, momentum, and studies of mantle dynamics, although in some studies energy are (see Chapter 7.02): surface velocities may be given in consistent with

ui;i ¼ 0 ½1 surface plate motions (e.g., Bunge et al., 1998). When steady-state or statistically steady-state solutions are ij ; j þ RaTi3 ¼ 0 ½2 to be sought, as they often are in mantle dynamics, qT the choice of initial condition can be rather arbitrary þ u T ¼ðT Þ þ ½3 qt i ;i ;i ;i and it does not significantly affect final results in a statistical sense. where ui, ij, T, and are the velocity, stress tensor, temperature, and heat-production rate, respectively, Although full time-dependent dynamics of ther- mal convection involves all three governing Ra is , and ij is a Kronecker delta function. Repeated indexes denote summation, and equations, an important subset of mantle dynamics problems, often termed as instantaneous Stokes flow u,i represents partial derivative of variable u with problem, only require solutions of eqns [1] and [2]. respect to coordinate xi. These equations were obtained by using the following characteristic scales: For Stokes flow problem, one may consider the length D, time D2/; and temperature T, where D dynamic effects of a given buoyancy field (e.g., one is often the thickness of the mantle or a fluid layer, derived from seismic structure) or prescribed surface is thermal diffusivity, and T is the temperature plate motion on gravity anomalies, deformation rate, difference across the fluid layer (see Chapters 7.02 and stress at the surface and the interior of the mantle and 7.04 for discussion on nondimensionalization). (Hager and O’Connell, 1981; Hager and Richards, 1989, also Chapters 7.02 and 7.04). The stress tensor ij can be related to strain rate "_ij see via the following constitutive equation: These governing equations generally require numerical solution procedures for three reasons. ¼ – P þ 2"_ ¼ – P þ ðu ; þ u ; Þ½4 ij ij ij ij i j j i (1) The advection of temperature in eqn [3], ui T;i , where P is the dynamic pressure and is the represents a nonlinear coupling between velocity and viscosity. temperature. (2) The constitutive law or eqn [4] is Substituting eqns [4] into [2] reveals three pri- often nonlinear in that stress and strain rate follow a mary unknown variables: pressure, velocity, and power-law relation; that is, the viscosity in eqn [4] temperature. The three governing eqns [1]–[3] are can only be considered as effective viscosity that sufficient to solve for these three unknowns, together depends on stress or strain rate. (3) Even for the with adequate boundary and initial conditions. Initial Stokes flow problem with a linear rheology, spatial conditions are only needed for temperature due to variability in viscosity can make any analytic solution the first-order derivative with respect to time in the method difficult and impractical. Numerical Methods for Mantle Convection 229

Irrespective of numerical methods used for the order accuracy were used to solve two-dimensional treatment of the individual governing equations, it (2-D) problems with variable viscosity (Christensen, is usual to solve the coupled system explicitly in time 1984). Malevsky (1996) developed a 3-D mantle con- as follows: (1) At a given time step, solve eqns [1] and vection code based on 3-D splines, which allowed [2] (i.e., the instantaneous Stokes flow problem) for one to reach very high Rayleigh numbers, like 108 flow velocity for given buoyancy or temperature. (Malevsky and Yuen, 1993). Recently, Kageyama and (2) Update the temperature to next time step from Sato (2004) have developed a 3-D FD technique, eqn [3], using the new velocity field. (3) Continue using a baseball-like topological configuration called this process of time stepping by going back to step (1). the ‘yin–yang’ grid, for solving the 3-D spherical convection problem. 7.05.3 Finite-Difference, Finite- Volume, and Spectral Methods 7.05.3.1.1 FD implementation of the governing equations In this section, we will briefly discuss how finite- For 2-D mantle convection within the Boussinesq difference (FD), finite-volume (FV), and spectral approximation and isoviscous flow in FD methods, methods are used in studies of mantle convection. a commonly used formulation employs stream-func- These methods have a long history in modeling tion and vorticity ! to eliminate the pressure and mantle convection (e.g., Machetel et al., 1986; Gable velocities, and the governing equations are written in et al., 1991), and remain important in the field (e.g., terms of stream function , vorticity !, and Tackley, 2000; Kageyama and Sato, 2004; Stemmer temperature T (e.g., McKenzie et al., 1974). They et al., 2006; Harder and Hansen, 2005). are written in time-dependent form as (also see eqns [1–3]) 7.05.3.1 Finite Difference r2 ¼ – ! ½8 FD methods have a much earlier historical beginning qT than FE, spectral, or FV methods because they are r2! ¼ Ra ½9 qx motivated intuitively by differential calculus and are qT q qT q qT based on local discretization of the derivative opera- ¼r2T – – þ ½10 tors based on a Taylor series expansion with an qt qx qz qz qx assigned order of accuracy about a given point. The where the velocity vector is defined as unknowns at each grid point depend on those of the neighboring points from the local Taylor expansion. q q v ¼ðv ; v Þ¼?ð ; – ?Þ½11 For examples of FD implementation, the reader is x z qz qx urged to consult an excellent introductory book by Lynch (2005). Additional material of FORTRAN 95 which automatically satisfies the continuity, and ! listings of library subprograms using FD methods can is the only component left of the vorticity vector be found in Griffiths and Smith (2006). Hence FD rv, x and z are the horizontal and vertical formulations are easy to grasp and to program and coordinates, respectively, with z vector pointing they have been the leading tool in numerical compu- upward, and t is the time. We note that eqns [8] and tations since the 1950s, as evidenced by the [9] are given by a set of coupled second-order partial pioneering codes written for meteorology and differential equations. Alternatively, we could have applied weapons research (e.g., Richtmeyer and combined them to form a single fourth-order partial Morton, 1967). All of the initial codes in mantle differential equation in terms of , called the bihar- convection were written with the FD formulation monic equation, as developed in the numerical (Torrance and Turcotte, 1971; Turcotte et al., 1973; scheme of Christensen (1984) using bicubic splines McKenzie et al., 1974; Parmentier et al., 1975; Jarvis and also by Schott and Schmeling (1998) using and McKenzie, 1980). The initial numerical techni- FDs. We note that solving the biharmonic solution ques in solving mantle convection in the nonlinear by FDs takes more time than solving two coupled regime were based on second-order FD methods Laplacian equations. We have restricted ourselves (Torrance and Turcotte, 1971; McKenzie et al., here to constant viscosity for pedagogical purposes. 1974). In the early 1980s, splines with a fourth- Examples of variable-viscosity convection equations 230 Numerical Methods for Mantle Convection can be found in Christensen and Yuen (1984) and We note that in FD and FV methods the primitive Schubert et al. (2000). variables are now dominant in terms of popularity in Zero tangential stress and impermeable boundary 3-D mantle convection with variable viscosity conditions can be expressed as (Tackley, 1994; Albers, 2000; Ogawa et al., 1991; Trompert and Hansen, 1996). !j ¼ j ¼ 0 ½12 Other boundary conditions are similar to those dis- cussed in Section 7.05.2. 7.05.3.1.2 Approximations of spatial In 2-D mantle convection problems, one can also derivatives and solution approaches utilize the so-called primitive variables (i.e., velocity In FD methods, spatial derivations in the above and dynamical pressure) formulation by solving the equations need to be approximated by values at grid coupled equations of mass and momentum conserva- points. There are many algorithms for generating the tions at each time step. This procedure may involve formulas in the FD approximation of spatial differ- more numerical care and computational time because ential operators (e.g., Kowalik and Murty, 1993; of more unknowns. The examples can be found in Lynch, 2005). The most common algorithm is to Auth and Harder (1999) and Gerya and Yuen (2003, use a Taylor expansion. For example, a second- 2007). order differential operator (for simplicity, consider For 3-D mantle convection, we can similarly the spatial derivative in the x-direction) can be employ the poloidal potential and a vorticity-like expressed as scalar function (Busse, 1989; Travis et al., 1990). It d2f ðxÞ is to be noted that this potential is not the same as – = 2 2 ¼½f ðx1Þþf ðx3Þ 2f ðx2Þ ð xÞ ½18 the stream function in 2-D (see Chapter 7.04). From dx the general representation of an arbitrary solenoidal where xk is a grid point while the derivative is com- vector field (Busse, 1989), we can write a 3-D velo- puted at a point x which is not necessarily a grid point city vector as (Figure 1(a)). This approximation is second-order accurate. We will use a recursive algorithm for gen- v ¼rrðezÞþrðezYÞ½13 erating the weights in the FD approximation with where ez is the unit vector in the vertical z-direc- high-order accuracy. We will, for simplicity, consider tion pointing upward. Y is the toroidal potential the spatial derivative in the x-direction. The nth- and is present in problems with lateral variations order differential operator can be approximated by of viscosity (Zhang and Christensen, 1993; Gable a FD operator as et al., 1991). Thus, for constant viscosity, Y is a n Xj constant and the velocity vector v ¼ ðu; v; wÞ d f ðxÞ n ¼ Wn;j ;kf ðxkÞ½19 involves higher-order derivatives of in this dx k ¼ 1 formulation: where Wn, j,k are the weights to be applied at grid q2 q2 q2 q2 point x , in order to calculate the nth order derivative u ¼ ; v ¼ ; w ¼ – þ ½14 k qyqz qxqz qx2 qy2 at the point x. The stencil is j grid points wide. An ideal algorithm should accurately and efficiently pro- The 3-D momentum equation for constant prop- duce weights for any order of approximation at erties can be written as a system of coupled Poisson arbitrarily distributed grid points. Such an algorithm equations in 3-D: has been developed and tested for simple differential r2 ¼ ½15 operators by Fornberg (1990, 1995). The cost for generating each weight is just four operations and 2 r ¼ RaT ½16 this can be done simply with a FORTRAN program with the energy equation as given in Fornberg’s book (Fornberg, 1995). The weights are collected in a ‘differentiation matrix’. qT ¼r2T – v ? rT þ ½17 This results in calculating the derivatives of a vector, qt which is derived from a matrix–vector multiplicative where all differential operators are 3-D in character, operation. This algorithm can be easily generalized and is a scalar function playing a role analogous to for determining the spatial derivatives for higher- the vorticity in the 2-D formulation. dimensional operators. Numerical Methods for Mantle Convection 231

(a) (b) Δx 123 W P E x x Δ Δ (δ (δ x x x)w x)e

(c) (d)

p p v u

(e)

v p u

Figure 1 A simple three-point one-dimensional (1-D) finite-difference stencil (a), a 1-D finite-volume control-cell (b), a staggered 2-D grid for velocity–pressure for horizontal (c) and vertical (d) components of the momentum equation and for the continuity equation (e).

In general, a pth-order FD method converges as conditions for the stream function and vorticity, u is O(k p) for k grid points. However, variable coeffi- the solution vector over the set of 2-D grid points for cients and nonlinearities, such as the heat advection ð; Þ, and f is a vector representing the right-hand term, greatly complicate the convergence. The side of the system from eqns [8] and [9], namely reader can find a more detailed discussion of the and qT=qx values at 2-D grid points from the pre- issues related to high-order FD methods and spectral vious time step. methods in Fornberg (1995). Mantle convection in The matrix equation from 2-D mantle convection both 2-D and 3-D configurations using this variable- problems with variable viscosity can be solved effec- grid, high-order FD method has been studied by tively and very accurately with the direct method, Larsen et al. (1995). such as the Cholesky decomposition, with current Using the stream-function vorticity formulation computational memory architecture. A high-resolu- for 2-D and the scalar potential approach for 3-D, tion 2-D problem with around 700 700 grid points we can see that we have a large system of algebraic and billions of tracers (Rudolf et al., 2004) can be equations to solve at each time step. The partial tackled on today’s shared-memory machines with differential equations posed by eqns [8] and [9] for 1–2 TB of RAM memory, such as the S.G.I. Altix at 2-D and eqns [15] and [16] in 3-D represent one of National Center for Supercomputing Applications the difficult aspects in mantle convection problems (NCSA) or the IBM Power-5 series in National because of its elliptic nature. The linear algebra aris- Center for Atmospheric Research (NCAR). This ing from this elliptic problem for large systems with tact of using shared-memory architecture has been varying size of matrix elements due to variable visc- employed by Gerya et al. (2006) to solve high-resolu- osity makes it very difficult to solve accurately. They tion 2-D convection problems with strongly variable can be written down symbolically as a large-scale viscosity. The presence of variable viscosity would matrix algebra problem, make the matrix A very singular because of the dis- parity of values to the matrix elements. The adverse Au ¼ f ½20 condition of the matrix is further aggravated by where A is a matrix operator involving the second- the presence of non-Newtonian rheology, which order spatial derivatives and the associated boundary makes the matrix A nonlinear, thus making it 232 Numerical Methods for Mantle Convection necessary to use iterative method for solving the temperature because this will lead closer to a matrix equation. Multigrid iterative method (e.g., pseudospectral quality. Other advection schemes Hackbush, 1985; Press et al., 1992; Trottenberg et al., include the semi-Lagrangian technique, which is 2001; Yavneh, 2006) is a well-proven way of solving based on tracer characteristics and have been used the elliptic problem with strong nonlinearities. in mantle convection for the stream-function Davies (1995) solved variable-viscosity convection method (Malevsky and Yuen, 1991) and for primi- problem in 2-D with the FD-based multigrid tive variables (Gerya and Yuen, 2003). technique. For 3-D mantle convection problems, one must still employ the iterative method because of memory 7.05.3.2 FV Method requirements associated with a FD grid configuration with over a million unknowns. The multigrid itera- A FV method is often used to solve differential equa- tive method used for an FD grid has been used by tions (Patankar, 1980). FV methods share some Parmentier and Sotin (2000) for solving 3-D high common features with FE and FD methods. In FV Rayleigh number mantle convection. We note that methods, discretization of a differential equation for variable-viscosity convection in 3-D one must use results from integrating the equation over a control- the pressure–velocity formulation cast in FDs (e.g., volume or control-cell and approximating differen- Peyret and Taylor, 1983) for the momentum equa- tial operators of reduced order at cell boundaries tion. The resulting FD equations can be solved with (Patankar, 1980). Patankar (1980) suggested that the the multigrid method. In this connection higher- FV formulation may be considered as a special case order FD method (Fornberg, 1995) with variable of the weighted residual method in FE in which the grid points together with the multigrid technique weighting function is uniformly one within an ele- may bring about a renaissance to the FD method ment and zero outside of the element. The FV because of its favorable posture in terms of memory method is also similar to FD method in that they requirements over FEs. both need to approximate differential operators using Another source of numerical difficulties in the values at grid points. mantle convection is due to the temperature advec- We will use a simple example to illustrate the tion term, v ? rT. It is well known from linear basic idea of the FV method (Patankar, 1980). analysis that numerical oscillations result, if simple Consider a 1-D heat conduction equation with heat FD schemes are used to calculate the spatial deri- source . vative due to the interaction between velocity and d dT k þ ¼ 0 ½21 the FD approximations of rT. Excellent discus- dx dx sions of this problem involving numerical dispersion can be found in Kowalik and Murty where k is the heat conductivity. This equation is (1993). Different numerical approximations have integrated over a control-cell bounded by dashed been proposed to treat the advection term in the lines in Figure 1(b), and the resulting equation is Z FD: the first is by Spalding (1972), which involves a dT dT e k – k þ dx ¼ 0 ½22 weighted upwind scheme and is first-order accurate, dx dx which has been employed in the early FD codes of e w w mantle convection (Turcotte et al., 1973). A more where e and w represent the two ends of the control- popular and effective method is an iterated upwind cell (Figure 1(b)). Introducing approximations for correction scheme that is correct to second order the flux at each end and the source term in eqn [22] and was proposed by Smolarkiewicz (1983, 1984). leads to This is a scheme based on the positively definite k ðT – T Þ k ðT – T Þ character of the advection term. The solutions, in e E P – w p W þ x ¼ 0 ½23 ðxÞ ðxÞ general, do not change much after one to two cor- e w rection steps and it is easy to program for parallel where ke and kw are the heat conductivity at cell computers. This scheme has been implemented by boundaries, TE, TP, and TW are temperatures at Parmentier and Sotin (2000) and by Tackley (2000) nodal points, is the averaged source in the con- for mantle convection problems. Use of higher- trol-cell, and x ¼ ½ðxÞe þ ðxÞw=2 is the size of order FD schemes (Fornberg, 1995) will also help the control-cell (Figure 1(b)). The eqn [23] is a to increase the accuracy of the advection of discrete equation with nodal temperatures as Numerical Methods for Mantle Convection 233 unknown for the given control-cell. Applying this Like in FE and FD methods, the FV method also procedure to all control-cells leads to a system of requires special treatment of advection term v ? rT equations with unknown temperatures at all grid in the energy equation. Either upwind scheme or an points, similar to eqn [20] from a FD method. iterative correction scheme such as that proposed by FV methods have also been used extensively in Smolarkiewicz (1983, 1984) in Section 7.05.3.1 on FD modeling mantle convection, possibly starting with can be used in the FV method to treat the advection Ogawa et al. (1991). Tackley (1994) implemented an term. FV method coupled with a multigrid solver for 2-D/3-D Cartesian models. Ratcliff et al. (1996) developed a 3-D spherical-shell model of mantle 7.05.3.3 Spectral Methods convection using an FV method. Harder and Hansen (2005) and Stemmer et al. (2006) also devel- The spectral method is a classical method, which is oped FV mantle convection codes for spherical- motivated by its analytical popularity and inherent shell geometries using a cubed sphere grid. All accuracy. It is based on the concept of orthogonal these mantle convection studies used second-order eigenfunction expansion based on the differential accurate FV methods, although higher-order for- operators associated with the Laplace equation and mulas can be formulated. works on orthogonal curvilinear coordinate systems. To solve fluid flow problems that are governed by For mantle convection problems this means that we the continuity and momentum equations, such as those can express the horizontal dependence by using in mantle convection, it is often convenient to use the Fourier expansion for Cartesian geometry and sphe- pressure–velocity formulation, similar to FE method. rical harmonics for the 3-D spherical shell. Also, a staggered grid is often used in which pressures Symbolically we can write down this expansion as and velocities are defined at different locations of a X control-cell (Patankar, 1980; Tackley, 1994) (Figures Fðx1; x2; x3Þ¼ aijkf ðx3Þgðx1Þhðx2Þ½24 1(c)–1(e)). Such a staggered grid helps remove check- ijk erboard pressure solutions. In the staggered grid, where F is the field variable being expanded, aijk is the control-cells for the momentum equation are different spectral coefficient, x1 and x2 are the horizontal coor- from those for the continuity equation. For example, dinates, x3 is the vertical or radial coordinate, g and h for the continuity equation, a pressure node is at the are the eigenfunctions being employed, and f is a center of a control-cell, while velocities are defined at function describing the vertical or radial dependence. the cell boundaries such that each velocity component We note that f can be determined by solving a two- is perpendicular to the corresponding cell boundary point boundary value, using propagator matrices, or (Figure 1(e)). For the momentum equation, the velo- an orthogonal function expansion, as in the case of cities are at the center of a cell, while the pressures are Chebychev polynomials, or using the FD (e.g., defined at cell boundaries (Figures 1(c)and1(d)). Cserepes and Rabionowicz, 1985; Cserepes et al., For a given pressure field, applying the FV pro- 1988; Gable et al., 1991; Travis et al., 1990; Machetel cedure leads to discrete equations for velocities. et al., 1986; Glatzmaier, 1988). However, solutions to the velocity equations are Balachandar and Yuen (1994) developed a spectral- only accurate if the pressure field is accurate. The transform 3-D Cartesian code for mantle convection, continuity equation is used to correct the pressure based on expansion of Chebychev functions for sol- field. This iterative procedure between the velocity ving the two-point boundary-value problem in the and pressure is implemented efficiently in a vertical direction and fast Fourier transforms along SIMPLER algorithm (Patankar, 1980) that is used the two horizontal directions. This method has also in the FV convection codes (Tackley, 1993, 1994; been extended to variable-viscosity problems with the Stemmer et al., 2006). A variety of methods can be help of Krylov subspace iterative technique (Saad and used to solve the discrete equations of velocities and Schultz, 1986) for solving iteratively the momentum pressure, similar to that in FD method. They may equation. The first 3-D codes in spherical-shell con- include successive over-relaxation and Gauss–Seidel vection were developed by Machetel et al.(1986)who iteration (Harder and Hansen, 2005; Stemmer et al., used an FD scheme in the radial direction and sphe- 2006), or a multigrid method (Tackley, 1994), or the rical harmonic decomposition over the longitudinal alternating-direction implicit (i.e., ADI) method and latitudinal directions and by Glatzmaier (1988) (Monnereau and Yuen , 2002). who employed Chebychev polynomials in solving 234 Numerical Methods for Mantle Convection the radial direction and spherical harmonics for the realistic mantle convection problems. Spectral meth- spherical surface. The Chebychev polynomials allow ods still remain the main driver for geodynamo for very high accuracy near the thermal boundary problems (e.g., Glatzmaier and Roberts, 1996; Kuang layers. They can also be expanded about internal and Bloxham, 1999) because of the exclusive choice of boundary layers, such as the one at 670 km constant material properties in solving the set of mag- discontinuity. Zhang and Yuen (1995 and 1996a) neto-hydrodynamic equations. Lastly, we wish to have devised a 3-D code for spherical geometry mention briefly a related spectral method called the based on higher-order FD method (Fornberg, 1995) spectral-element method (Maday and Patera, 1989), and spectral expansion in the field variables in which is now being used in in seismic- spherical harmonics along the circumferential direc- wave propagation (Komatitsch and Tromp, 1999). tions. They also developed an iterative technique for This approach may hold some promise for variable- the momentum equation to solve for variable viscosity viscosity mantle convection. with a lateral viscosity contrast up to 200 (Zhang and Yuen, 1996b) in compressible convection. It is impor- tant to check also the spectra of the viscosity decay 7.05.4 An FE Method with the degree of the spherical harmonics (Zhang and Yuen, 1996a) in order to ascertain whether Gibbs FE methods are effective in solving differential equa- phenomenon is present. tions with complicated geometry and material Spectral methods are very accurate, much more so properties. FE methods have been widely used in than FEs and FDs for the same computational efforts, the studies of mantle dynamics (Christensen, 1984; and their efficacy in terms of convergence can be Baumgardner, 1985; King et al., 1990; van den Berg assessed by examining the rate of decay of the energy et al., 1993; Moresi and Gurnis, 1996; Bunge et al., in the spectrum with increasing number of terms in 1997; Zhong et al., 2000). This section will go through the spectral expansion in eqn [24] (Peyret and Taylor, some of the basic steps in using FE methods in sol- 1983). Spectral methods also have a couple of other ving governing equations for thermal convection. distinct advantages. First, they are very easy to imple- The FE formulation for Stokes flow that is ment, because they reduce the problems to either an described by eqns [1] and [2] is independent from algebraic set or simple weakly coupled ordinary dif- that for the energy equation. Hughes (2000) gave ferential equations, as discussed earlier. Second, using detailed description on a Galerkin weak-form fast transform algorithms such as fast Fourier trans- FE formulation for the Stokes flow. Brooks (1981) form, spectral methods can be very efficient and fast developed a streamline upwind Petrov–Galerkin for- on a single-processor computer. mulation (SUPG) for the energy equation involving However, spectral methods only work well for advection and diffusion. These two formulations constant material properties or depth-dependent remain popular for solving these types of problems properties in simple geometries (Balachandar and (Hughes, 2000) and are employed in mantle convec- Yuen, 1994). Furthermore, spectral techniques do not tion codes ConMan (King et al., 1990) and Citcom/ go as far in terms of viscosity contrasts (less than a CitcomS (Moresi and Gurnis, 1996; Zhong et al., factor of 200) for variable-viscosity convection 2000). The descriptions presented here are tailored (Balachandar et al., 1995, Zhang and Yuen, 1996b). from Brooks (1981), Hughes (2000), and Ramage and We note, however, that Schmalholz et al.(2001) Wathen (1994) specifically for thermal convection in showed that in a folding problem their spectral an incompressible medium, and they are also closely method could go up to a viscosity contrast of around related to codes ConMan and Citcom. 104. Perhaps a better choice of a preconditioner in the solution of the momentum equation in the spectral 7.05.4.1 Stokes Flow: A Weak Formulation, expansion of a variable-viscosity problem will enable a Its FE Implementation, and Solution larger viscosity contrast than 104. This issue is still open and remains a viable research topic. Spectral 7.05.4.1.1 A weak formulation methods are also difficult to be used efficiently on The Galerkin weak formulation for the Stokes flow parallel computers because of the global basis func- can be stated as follows: find the flow velocity tions used in these methods. At present, spectral uj ¼ vi þ gi and pressure P, where gi is the prescribed methods, although elegant mathematically, are not as boundary velocity from eqn [6] and vi PV, and PPP, effective as other numerical methods for solving where V is a set of functions in which each function, Numerical Methods for Mantle Convection 235

wi, is equal to zero on gi , and P is a set of functions q, such that for all wi PV and qPP Z Z Z

wi; j ij d – qui; i d ¼ wi fi d Z Xnsd þ w h d ½25 i i v i¼1 hi wi and q are also called weighting functions. Equation p [25] is equivalent to eqns [1] and [2] and boundary Figure 2 Finite-element discretization and grid in 2-D. For a conditions equation [6], provided that fi ¼ RaTiz (Hughes, 2000). Equation [25] can be written as four-node bilinear element, the pressure is defined at the center Z Z Z of the element, while velocities are defined at the four corners. w c v d– qv d– w P d i;j ijkl k;l i;i i;i v p Z Z Z is the set of velocity nodes, is the set of Xnsd pressure nodes, and v is the set of velocity nodes ¼ wi fi d þ wi hi d– wi;j cijkl gk;l d ½26 gi h i¼1 i along boundary gi . Note that the velocity shape where functions and velocity nodes can be (and usually are) different from those for pressure (Figure 2). cijkl ¼ ðikjl þ il jkÞ½27 Substituting eqns [30] into [28] leads to the fol- is derived from constitutive eqn [4]. It is often con- lowing equation: venient to rewrite "ðwÞTD"ðvÞ 0 1 0 1 T T wi; j cijkl vk; l ¼ "ðwÞ D"ðvÞ½28 X X @ A @ A where for 2-D plane strain problems, ¼ " NAwiAei D" NBvj Bej APv – v BPv – v 8 9 2 3 gi gj > > " # " # <> v1;1 => 2 00 X X 6 7 T 6 7 ¼ "ðNAei Þ wiA D "ðNBej Þvj B "ðvÞ¼ v2;2 ; D ¼ 4 02 0 5 ½29 > > APv – v BPv – v : ; gi " gj # v1;2 þ v2;1 00 X X ¼ w eTBTDB e v ½32 (note here the change in the definition of the off- iA i A B j j B APv – v BPv – v diagonal components of the strain-rate tensor). It is gi gj where for 2-D plane strain problems straightforward to write the expressions for other 2 3 coordinate systems including 3-D Cartesian, axisym- NA;1 0 6 7 metric (Hughes, 2000), or spherical geometry (Zhong 6 7 BA ¼ 4 0 NA;2 5 ½33 et al., 2000). Suppose that we discretize the solution domain NA;2 NA;1 using a set of grid points (Figure 2) so that the velocity Substituting eqns [30] and [31] into [26] leads to the and pressure fields and their weighting functions can following equation: " ! be expressed anywhere in the domain in terms of their X X Z values at the grid (nodal) points and the so-called T T wiA ei BA DBB d ej vj B ‘shape functions’ which interpolate the grid points: APv – v BPv – v X gi gj !# X Z v ¼ vi ei ¼ NAviAei ; P v v – ei NA; i MB d PB A – g X i BPp " Z !# w ¼ wi ei ¼ NAwiAei ; X X P v v A – g – q M N d e v X i A A B; j j j B APp BPv – v g ¼ N g e ½30 gj A iA i " Z Z APv X Xnsd gi X X ¼ wiA NAei fi d þ NAei hi d P v v h ; A – g i¼1 i P ¼ MBPB q ¼ MBqB ½31 i !# BPp BPp X Z T T – e B DBB d ej gj B ½34 where NA is the shape functions for velocity at node i A BPv gj A, MB is the shape functions for pressure at node B, 236 Numerical Methods for Mantle Convection

Because eqn [34] holds for any weighting func- of integrals over the solution domain and can, tions wiA and qA, it implies the following two therefore, without approximation, be written as a equations: sum of integrals over convenient subdomains, and Z ! the matrix eqn [37] may be decomposed into X overlapping sums of contributions from these eT BTDB d e v i A B j jB domains. BPv – v gj ! Let us first introduce the elements and shape X Z functions. A key feature of the standard FE method – ei NA; i MB d PB P p is that a local basis function or shape function is Z B Z Xnsd used such that the value of a variable within an ¼ NAei fi d þ NAei hi d element depends only on that at nodal points of h i¼1 i ! the element. The shape functions are generally X Z T T chosen such that they have the value of unity on – ei BA DBB d ej gj B ½35 BPv their parent node and zero at all other nodes and gj zero outside the boundary of the element. Unless Z ! X there is a special, known form to the solution, it is MANB; j d ej vj B ¼ 0 ½36 usual to choose simple polynomial shape functions BPv – v gj and form their products for additional dimensions. In addition, this interpolating requirement con- Combining eqns [35] and [36] into a matrix form strains the patterns of nodes in an element and leads to which elements can be placed adjacent to each "#() () other. Bathe (1996) gives an excellent overview of KG V F the issues. (As usual, there are some useful excep- ¼ ½37 GT 0 P 0 tions to this rule, one being the element-free Galerkin method where smooth, overlapping inter- where the vector V contains the velocity at all the polating kernels are used. This does mean, nodal points, the vector P is the pressure at all the however, that the global problem cannot be trivi- pressure nodes, the vector F is the total force term ally decomposed into local elements.) Zienkiewicz resulting from the three terms on the right-hand side et al. (2005) is also an excellent source of reference of eqn [35], the matrices K, G, and GT are the stiffness for these topics. matrix, discrete gradient operator, and discrete For simplicity, we consider a 2-D domain with divergence operator, respectively, which are derived quadrilateral elements. We employ mixed elements from the first and second terms of eqns [35] and in which there are four velocity nodes per element [36], respectively. Specifically, the stiffness matrix is each of which occupies a corner of the element, while given by the only pressure node is at the center of the element Z (Figure 2). For these quadrilateral elements, the T T velocity interpolation in each element uses bilinear Klm ¼ ei BA DBB d ej ½38 shape functions, while the pressure is constant for where subscripts A and B are the global velocity node each element. numbers as in eqn [30], i and j are the degree of As a general remark on FE modeling of deforma- tion/flow of incompressible media, it is important to freedom numbers ranging from 1 to nsd, and l and m are the global equation numbers for the velocity keep interpolation functions (shape functions) for velocities at least 1 order higher than those for pres- ranging from 1 to nvnsd where nv is the number of velocity nodes. sure, as we did for our quadrilateral elements (Hughes, 2000). Spurious flow solutions may arise sometimes even if this condition is satisfied. The 7.05.4.1.2 An FE implementation best-known example is the ‘mesh locking’ that arises We now present an FE implementation of the from linear triangle elements with constant pressure Galerkin weak formulation for the Stokes flow per element for which incompressibility (i.e., a fixed and the resulting expressions of different terms in elemental area) constraint per element demands zero [37]. A key point of the FE approach is that all of deformation/flow everywhere in the domain the equations to be solved are written in the form (Hughes, 2000). Numerical Methods for Mantle Convection 237

For any given element e, velocity and pressure use isoparametric elements for which the coordinates within this element can be expressed through the and velocities in an element have the interpolation following interpolation: schemes (Hughes, 2000). For example, for 2-D quad- rilateral elements that we discussed earlier, the Xn Xn en en velocity shape functions for node a of an element in v ¼ vi ei ¼ Naviaei ; w ¼ wi ei ¼ Nbwibei ; P a¼1 a parent domain with coordinates ð – 1; 1Þ and b¼1 P Xnen ð – 1; 1Þ is given as g ¼ gi ei ¼ Nagiaei ½39 a¼1 Nað; Þ¼1=4ð1 þ aÞð1 þ aÞ½45 Xnep Xnep ; P ¼ MaPa q ¼ Maqa ½40 where ða; aÞ is (1,1), (1, 1), (1, 1), and (1, 1) a¼1 a¼1 for a ¼ 1, 2, 3, and 4, respectively. The pressure where nen and nep are the numbers of velocity and shape functions for Ma is 1, as there is only one pressure nodes per element, respectively, and nen ¼ 4 pressure node per element. Although the integrations and nep ¼ 1 for our quadrilateral elements. The shape in [42]–[44] are in the physical domain (i.e., x1 and x2 function Na for a ¼ 1, ...,nen depends on coordinates, coordinates) rather than the parent domain, they can and Na is 1 at node a and linearly decreases to zero at be expressed in the parent domain through coordi- other nodes of the element. The locality of the shape nate transformation. functions greatly simplifies implementation and In practice, these element integrals are calculated computational aspects of the Galerkin weak formula- numerically by some form of quadrature rule. In 2-D, tion. For example, the integrals in eqns [35] and [36] Gaussian quadrature rules are optimal and are may be decomposed into sum of integrals from each usually recommended; in 3-D, other rules may be element and the matrix equation [37] may be decom- more efficient but are not commonly used for reasons posed into sums of elemental contributions. of programming simplicity (Hughes (2000) docu- Specifically, we may now introduce elemental stiff- ments integration procedure in detail). There are a ness matrix, discrete gradient and divergence small number of cases where it may be worthwhile operators, and force term. using a nonstandard procedure to integrate an ele- ment. The most common is where the constitutive e e e e e e k ¼½klm; g ¼½gln; f ¼ffl g½41 parameters (i.e., D in eqn [29]) change within the element; a higher-order quadrature scheme than the where 1 l; m nennsd; 1 n nep (note that for standard recommended one can give improved accu- quadrilateral elements, nen ¼ 4, nep ¼ 1, and nsd ¼ 2), racy in computing ke. When the constitutive e e k is a square matrix of nennsd by nennsd , and g is a parameters are strongly history dependent, D is matrix of nennsd by nep. known only at a number of Lagrangian sample points; Z if these are used directly to integrate ke,the e T T klm ¼ ei Ba DBb d ej ½42 Lagrangian integration point methods such as MPM e result (Sulsky et al., 1994; Moresi et al., 2003 for application in ). where l ¼ nsd(a 1) þ i, m ¼ nsd(b 1) þ j, a,b ¼ With elemental ke, ge, and f e determined, it is 1, ...,nen, and i,j ¼ 1, ...,nsd Z straightforward to assemble them into global matrix e eqn [37]. If an iterative solution method is used to gln ¼ – ei Na; i Mn d ½43 e solve [37], one may carry out calculations of the left- hand side of [37] element by element without assem- where n ¼ 1, ...,n , and the rest of the symbols have ep bling elemental matrices and force terms into the the same definitions as before: global matrix equation form [37]. Z Z nXsdnen The boundary conditions described in eqn [44] e e e fl ¼ Nafi d þ Nahi d – klmgm ½44 e are simple to implement when the boundary is e m¼1 hi aligned with the coordinate system, and when the Determinations of these elemental matrices and boundary condition is purely velocity or purely force term require evaluations of integrals over each boundary tractions. The difficulty arises in the case element with integrands that involve the shape func- of boundaries which are not aligned with the tions and their derivatives. It is often convenient to coordinate system and which specify a mixture of 238 Numerical Methods for Mantle Convection boundary tractions and velocities. This form of Iterative solution approaches are the only feasible boundary condition is common in engineering appli- and practical approaches for 3-D problems. Later cations of FE method, designing mechanical we will briefly discuss a penalty formulation for the components, for example, and in geodynamics it incompressible Stokes flow that requires a direct occurs when modeling arbitrarily oriented contact solution approach and is only effective for 2-D surfaces such as faults, or a spherical boundary in a problems. Cartesian solution domain. Unless treated appropri- The system of equations as it stands is singular ately, this form of boundary condition leads to a due to the block of zero entries in the diagonal, but it constraint on a linear combination of the degrees of is symmetric, and the stiffness matrix K is symmetric freedom at each constrained node. The solution is positive definite and we can use this to our advantage straightforward: instead of using the global coordi- in finding a solution strategy. An efficient method is nate system to form the matrix equation, we first the Uzawa algorithm which is implemented in transform to a local coordinate system which does Citcom code (Moresi and Solomatov, 1995). In the conform to the boundary geometry (Desai and Abel, Uzawa algorithm, the matrix equation [37] is broken 1972). into two coupled systems of equations (Atanga and Let R be a rotation matrix which transforms the Silvester, 1992; Ramage and Wathen, 1994): global coordinate system XYZO to a local one X9Y9Z9O9 aligned with the boundary at node n. KV þ GP ¼ F ½48

The transformed stiffness matrix and force vector T contributions for this node then become G V ¼ 0 ½49 Combining these two equations and eliminating V K 9 ¼ RTK R ½46 n n n n to form the Schur complement system for pressure and (Hughes, 2000)

T T – 1 T – 1 F9n ¼ Rn Fn ½47 ðG K GÞP ¼ G K F ½50 R can be formulated for each node or it may be Notice that matrix Kˆ ¼ GT K – 1G is symmetric posi- assembled for an element or for the global matrices tive definite. Although in practice eqn [51] cannot be but, other than the nodes where the ‘skew’ boundary directly used to solve for P due to difficulties in conditions occur, the block entries are simply the obtaining K1, we may use it to build a pressure- identity matrix. It is possible to use this procedure correction approach by using a conjugate gradient at every node in a system with a natural coordinate algorithm which does not require construction of system (e.g., the spherical domain) so that one can matrix Kˆ (Ramage and Wathen, 1994). The proce- switch freely between a spherical description of dure is presented and discussed in detail as follows. forces and velocities where convenient, and an With the conjugate gradient algorithm, for sym- underlying Cartesian formulation of the constitutive metric positive definite Kˆ, the solution to a linear relations. This method was used in modeling faults in system of equations KPˆ ¼ H can be obtained with Zhong et al. (1998). the operations in the left column of Figure 3 (Golub and van Loan, 1989, p. 523). 7.05.4.1.3 The Uzawa algorithm for the For equations [48] and [50] for both velocities and matrix equation pressure, with initial guess pressure P0 ¼ 0, the initial Similar to FD or FV methods, the FE discretization velocity V0 can be obtained from of the differential equations leads to a matrix equa- KV ¼ F; or V ¼ K – 1F ½51 tion such as [37]. The remaining question is to solve 0 0 the matrix equation, which we now discuss in this and the initial residual for pressure equation [50], r0, T – 1 section. We can use either direct or iterative methods and search direction, s1, are r0 ¼ s1 ¼ H ¼ G K F T to solve the matrix equation, depending on problems ¼ G V0 (see the left column of Figure 3). To deter- that we are interested, similar to what we discussed mine the search step k in the conjugate gradient earlier for matrix equation [20] from FD method. algorithm, we need to compute the product of search ˆ T ˆ Here we will focus on iterative solution methods, direction sk and K, sk Ksk (Figure 3). This product can because they require significantly less memory and be evaluated without explicitly constructing Kˆ for the computation than direct solution approaches. following reasons. Numerical Methods for Mantle Convection 239

k = 0; P0 = 0 Solve KV0 = F k = 0; P0 = 0; r0 = H T ε r0 = H = G V0 while |r k| > a given tolerance, while |r | > a given tolerance, ε k = k + 1 k if k = 1 k = k + 1 s1 = r 0 if k = 1 else s1 = r 0 T T else β = r k rk –1rk –1/rk –2 k –2 β = T T r = β r r /r k –2 sk rk –1 + k sk –1 k k –1 k –1 k –2 = β end sk rk –1 + k sk –1 α = T T ˆ k r k –1rk –1/sk Ksk end = α Pk Pk –1 + k sk Solve Kuk = Gsk T T α = r r /(Gs ) u r = r + α Ksˆ k k –1 k –1 k k k k –1 k k = α end Pk Pk –1 + k sk = α P = Pk Vk Vk –1 – k uk = α T rk rk –1 – k G uk end P = Pk, V = Vk Figure 3 The left column is the original conjugate gradient algorithm, and the right column is the Uzawa algorithm for solving eqns [48] and [49].

The product can be written as Pk and Vk into [48] and considering Pk ¼ Pk – 1 þ ksk, Vk ¼ Vk – 1 þ v, and [56] lead to T T T – 1 T – 1 s Ksˆ k ¼ s G K Gsk ¼ðGskÞ K Gsk ½52 k k – 1 Kv þ kGsk ¼ 0orv ¼ – kK Gsk ½57 If we define uk, such that From [53], it is clear that the velocity increment – 1 Kuk ¼ Gsk; or uk ¼ K Gsk ½53 v ¼ – kuk, and consequently eqn [55] updates the velocity. then we have The final algorithm is given in the right column of T T Figure 3 (Ramage and Wathen, 1994). The effi- sTKsˆ ¼ðGs Þ K – 1Gs ¼ðGs Þ u ½54 k k k k k k ciency of this algorithm depends on how efficiently This indicates that if we solve [53] for uk with Gsk eqn [53] is solved. We note that the choice of con- T ˆ as the force term, the product sk Ksk can be obtained jugate gradient is simply one of a number of possible without actually forming Kˆ. Similarly, Ksˆ k in updating choices here. Any of the ‘standard’ Krylov subspace the residual rk (the left column of Figure 3) can be methods including biconjugate gradient, GMRES, obtained without forming Kˆ, because can, in principle, be developed in the same way using [52]–[54] wherever matrix-vector products ˆ T – 1 T Ksk ¼ G K Gsk ¼ G uk involving the inverse of K are required.

As the pressure P is updated via Pk ¼ Pk – 1 þ ksk Preconditioning for the pressure equation can be of from the conjugate gradient algorithm (Figure 3), great help in improving the convergence of this itera- the velocity field can also be updated accordingly tion, as discussed in Moresi and Solomatov (1995). via It should be pointed out that the Uzawa algorithm outlined above can also be used in connection with Vk ¼ Vk – 1 – kuk ½55 other numerical methods including FV and FD This can be seen from the following derivation. At methods, provided that the matrix equations for pres- iteration step k 1, the pressure and velocity are sure and velocities from these numerical methods have the form of [37]. Pk – 1 and Vk – 1, respectively, and they satisfy eqn [48]

KV þ GP ¼ F ½56 k – 1 k – 1 7.05.4.1.4 Multigrid solution strategies At iteration step k, the updated pressure is Pk, The stiffness matrix K is symmetric positive definite, and the velocity Vk ¼ Vk – 1 þ v, where v is the and this allows for numerous possible solution unknown increment to be determined. Substituting approaches. For example, multigrid solvers have 240 Numerical Methods for Mantle Convection been used for solving eqn [53] (Moresi and For an elliptic operator such as the stiffness Solomatov, 1995) as in Citcom code. Newer versions matrix, K, of the Stokes flow problem in eqn [53] of Citcom including CitcomS/CitcomCU employ we write full multigrid solvers with a consistent projection K v ¼ F ½58 scheme for eqn [53] and are even more efficient h h h (Zhong et al., 2000). where the h subscript indicates that the problem has The multigrid method works by formulating been discretized to a mesh of fineness h. An initial the FE problem on a number of different scales – estimate of the velocity on grid h, vh, can be improved usually a set of grids which are nested one within by a correction vh determining the solution to the other sharing common nodes (see Figure 4(a)), K v ¼ F – K v ½59 similar to how multigrid methods are used in FD h h h h h and FV methods. The solution progresses on all of Suppose we obtain our approximate initial esti- the grids at the same time with each grid eliminating mate by solving the problem on a coarse grid. The errors at a different scale. The effect is to propagate reduction of the number of degrees of freedom leads information very rapidly between different nodes in to a more manageable problem which can be solved the grid which would otherwise be prevented by the quickly. The correction term is therefore local support of the element shape functions. In K v ¼ F – K RH v ½60 fact, by a single traverse from fine to coarse grid h h h h h H and back, all nodes in the mesh can be directly con- where H indicates a coarser level of discretization, H nected to every other – allowing nodes which are and the operator Rh is an interpolation from the physically coupled but remote in the mesh to com- coarse-level H to the fine-level h. municate directly during each iteration cycle. This To find vH we need to solve a coarse approxima- matches the physical structure of the Stokes flow tion to the problem problem in which stresses are transmitted instanta- K v ¼ F ½61 neously to all parts of the system in response to H H H changes anywhere in the buoyancy forces or bound- where KH and FH are the coarse-level equivalent of ary conditions. the stiffness matrix and force vector. One obvious The multigrid effect relies upon using an iterative way to define these is to construct them from a coarse solver on each of many nested grid resolutions which representation of the problem on the mesh H exactly acts like a smoother on the residual error at the as would be done on h. An alternative is to define characteristic scale of that particular grid. Gauss– K ¼ Rh K RH and F ¼ Rh F ½62 Seidel iteration is a very common choice because it H H h h H H h H has exactly this property, although its effectiveness where Rh is a ‘restriction’ operator which has the depends on the order in which degrees of freedom opposite effect to the interpolation operation in that are visited by the solver and, consequently, it can be it lumps nodal contributions from the fine mesh onto difficult to implement in parallel. On the coarsest the coarse mesh. grid it is possible to use a direct solver because the The power of the algorithm is in a recursive number of elements is usually very small. application. The coarse-grid correction is also

(a)

H (b) (c) Fine Fine

Coarse Coarse h iteration iteration

Figure 4 A nested grids with fine and coarse meshes (a), structure of V cycle multigrid iteration (b) and structure of a full multigrid Iteration (c). Numerical Methods for Mantle Convection 241

Obtain an approximate solution vh at the finest grid, h the residual on a number of grids, the whole problem = − is defined for all the grids. In this way the initial fine- Calculate residual rh Fh Khvh grid approximation is obtained by interpolating from Project residual by N levels to h – N the solution to the coarsest-grid problem. The solu- repeat: r = R h−(i −1)r h− i h− i h− i −1 tion at each level is still obtained by projecting to the Solve exactly finest level and reducing the residual at each projec- Δ = vh− N Kh− N rh− N tion step. The resulting cycle is illustrated in Interpolate and improve Figure 4(c). − A recent overview of multigrid methods is r = R h i K Δv h−(i −1) h−(i −1) h− i h− i required reading at this point. Yavneh (2006) intro- Improve Δv h−(i −1) duces the method which is then explained for a = +Δ vh−(i −1) vh−(i −1) vh−(i −1) number of relevant examples by Oosterlee and Figure 5 A simple sawtooth multigrid algorithm for solving Gaspar-Lorenz (2006), Bergen et al. (2006), and eqn [51]. Bastian and Wieners (2006). calculated through the use of a still coarser grid and 7.05.4.2 Stokes Flow: A Penalty so on, until the problem is so small that an exact Formulation solution can be obtained very rapidly. One very simple, but instructive, algorithm for hierarchical For 2-D problems, an efficient method to solve the residual reduction is the sawtooth cycle given in incompressible Stokes flow is a penalty formulation Figure 5, and its logical layout is same as the V- with a reduced and selective integration. This cycle in Figure 4(b). method has been widely used in 2-D thermal con- The step in which the velocity correction is vection problems, for example, in ConMan (King ‘improved’ is an iterative method for reducing the et al., 1990). We now briefly discuss this penalty residual at the current level which has the property of formulation, and detailed descriptions can be found smoothing the error strongly at the current mesh in Hughes (2000). scale. At each level these smoothing operators reduce The key feature in the penalty formulation is to the residual most strongly on the scale of the discre- allow for slight compressibility or uk; k 0. Here it is tization – the hierarchical nesting of different mesh helpful to make an analogy to isotropic elasticity. sizes allows the residual to be reduced at each scale The constitutive equations for both compressible very efficiently (see Yavneh (2006) for a more and incompressible isotropic elasticity are given by lengthy discussion). the following two equations: The projection and interpolation operators have to be chosen fairly carefully to avoid poor approx- ij ¼ – Pij þ ðui; j þ uj ; i Þ½63 imations to the problem at the coarse levels and u þ P= ¼ 0 ½64 ineffectual corrections propagated to the fine levels. k; k The interpolation operator is defined naturally from where is the Lame constant which is finite for the shape functions at the coarse levels. The projec- compressible media but infinite for incompressibility tion operator is then defined to complement this media (i.e., to satisfy uk; k ¼ 0 for finite P). To allow choice (the operators should be adjoint). for slight compressibility, is taken finite but signif- The sawtooth cycle outlined in this section is the icantly larger than , such that the error associated simplest multigrid algorithm. Developments include with the slight compressibility is negligibly small. improving the residual at each level of the restriction Using words of 64 bit long (i.e., double precision), as well as the interpolation, known as a ‘v-cycle’, and / 107 is effective. For finite , the constitutive cycles in which the residual is interpolated only part equation becomes way through the hierarchy before being reprojected ¼ u þ ðu þ u Þ½65 and subjected to another set of improvements (a ‘w- ij k; k ij i; j j ; i cycle’). which replaces eqn [4]. The full multigrid algorithm introduces a further An interesting consequence of this new constitu- level of complexity. Instead of simply casting the tive equation is that the pressure is no longer needed problem at a single level and projecting/improving in the momentum equation, and this simplifies the 242 Numerical Methods for Mantle Convection

FE analysis. The weak form of the resulting Stokes First, although the pressure is not directly flow problem is solved from the matrix equation, the pressure can be obtained through postprocessing via P ¼ – Z Z Z Xn sd uk; k for each element. Such obtained pressure fields wi; j cijkl vk; l d ¼ wi fi d þ wi hi d often display a checkerboard pattern. However, a i¼1 hi Z pressure-smoothing scheme (Hughes, 2000) seems – wi; j cijkl gk; l d ½66 to work well. The pressure field is important in many geophysical applications including computing where and melt migration. Second, with / 107, the stiffness matrix is not cijkl ¼ ij kl þ ð ik jl þ il jkÞ½67 well conditioned and is not suited for any iterative The FE implementation of eqn [66] is similar solvers. A direct solver is required for this type of to that in Section 7.05.4.1. With the pressure equations, as done in ConMan. This implies that this excluded as a primary variable, the matrix equation formulation may not be applicable to 3-D problems is simply because of the memory and computation require- ments associated with direct solvers. Reducing / ½K fV g¼fFg½68 improves the condition for the stiffness matrix; how- While the elemental force vector is defined the ever, this is not recommended as it results in large same as that in [44], the elemental stiffness needs errors associated with relaxing the incompressibility some modification in comparison with that in [42]: constraints. Z Z ! e T T T klm ¼ ei Ba DBb d þ Ba DBb d ej ½69 e e 7.05.4.3 The SUPG Formulation for the where the first integral is the same as in [42] but the Energy Equation second integral is a new addition with The convective transport of any quantity at 2 3 0 high Peclet number (the ratio of advective trans- 6 7 port rate to diffusion rate) is challenging in any D ¼ 6 0 7 ½70 4 5 numerical approach in which the grid does not 000 move with the material deformation. The transfer The matrix eqn [68] only yields correct solution of quantities from grid points to integration points for velocities if a reduced and selective integration in order to calculate their updated values intro- scheme is used to evaluate the elemental stiffness duces a nonphysical additional diffusion term. matrix (e.g., Hughes, 2000). Specifically, the Furthermore, the advection operator is difficult to numerical quadrature scheme for the second inte- stabilize and many different schemes have been gral of eqn [69] needs to be one order lower than proposed to treat grid-based advection in both an that used for the first integral. For example, if for a accurate and stable fashion. Many of the successful 2-D problem, a 2 2 Gaussian quadrature scheme approaches include some attempt to track the flow is used to evaluate the first integral, then a one- direction and recognize that the advection operator point Gaussian quadrature scheme is needed for is not symmetric in the upstream/downstream the second integral. Hughes (2000) discussed the directions. This section introduces an SUPG for- equivalence theorem for the mixed elements and mulation and a predictor–multicorrector explicit the penalty formulation with the reduced and algorithm for time-dependent energy equation selective integration. Moresi et al. (1996) showed (i.e., eqn [3]). This method was developed by that these two formulations yield essentially iden- Hughes (2000) and Brooks (1981) some twenty tical results for the Stokes flow problems by years ago and remains an effective method in FE comparing solutions from ConMan code employing solutions of the equations with advection and dif- a penalty formulation and Citcom which uses a fusion such as our energy equation. FE mantle mixed formulation. convection codes Citcom and ConMan both Finally, we make two remarks about this penalty employ this method for solving the energy formulation. equation. Numerical Methods for Mantle Convection 243

A weak-form formulation for the energy equation A reasonable assumption is the weighted diffusion [3] and boundary conditions [7] is (Brooks, 1981) for an element in the third term of eqn [71],

Z Z wðT;i Þ;i , is negligibly small. Therefore, eqn [71] _ can be written as wðT þ ui T;i Þd þ w;i ð T;i Þd Z Z Z X X _ _ w;i ðT;i Þd þ w˜ðT þ ui T;i – Þd þ w½T þ ui T;i – ðT;i Þ – d ;i Z Z e e Z e e Z Z ¼ wq d – w;i g;i d ½76 ¼ w d þ wq d – w; g; d ½71 i i q q We now present relevant matrices at an element where w is the regular weighting functions and is zero level. The T_ term in [71] implies that a mass matrix _ on Dq, T is the time derivative of temperature, and w is needed and it is given as is the streamline upwind contribution to the weight- Z ing functions. e mab ¼ NaNb d ½77 The FE implementation of [71] is similar to what e was discussed for the Stokes flow in Section 7.05.4.1.2. e where a, b ¼ 1,...,nen. Elemental stiffness k is While the weighting function w is similar to what was Z defined in [39] except it is now a scalar, the stream- e T line upwind part w is defined through artificial kab ¼ Ba Bb d ½78 e diffusivity ˜ as where for 2-D problems

w ¼ ˜uˆj w;j =jju ½72 T Ba ¼ðNa;1 Na;2Þ½79 Elemental force vector f e is given as where jju is the magnitude of flow velocity, uˆj ¼ uj = jju represents the directions of flow velocity, and ˜ is Z Z Xnen defined as e ˜ ˜ e e fa ¼ Na d þ Naq d – kabgb ½80 e e ! q b ¼ 1 Xnsd ˜ ˜ ¼ i ui hi 2 ½73 i¼1 where N˜a is the Petrov–Galerkin shape function. 8 Elemental advection matrix ce is given as > – – = ;< – <> 1 1 i i 1 Z ui hi ˜ e i ¼ 0; – 1 i 1 ; for i ¼ ½74 ˜ > 2 cab ¼ Naui Nb; i d ½81 : e 1 – 1=i; i > 1 The combined matrix equation may be written as

M_ þðK þ CÞ ¼ F ½82 where ui and hi are flow velocity and element lengths in certain directions. It should be pointed out that eqns where is the unknown temperature, and M, K, C, [72] and [74] are empirical and other forms are possi- and F are the total mass, stiffness, advection matrices, ble. Such defined streamline upward weighting and force vector assembled from all the elements. function w can be thought as adding artificial diffusion Equation [82] can be solved using a predictor– to the actual diffusion term to lead to total diffusivity corrector algorithm (Hughes, 2000) with some initial condition for temperature (e.g., eqn [5]). Suppose that þ ˜uˆ uˆ ½75 i j temperature and its time derivative at time step n are given, and _ , the solutions at time step n þ 1 w is discontinuous across elemental boundaries, dif- n n with time increment t can be obtained with the ferent from w. This is why the integral in the third following algorithm: term of [72] is for each element. w˜ ¼ w þ w is also sometimes called the Petrov–Galerkin weighting 1. Predictor: functions which indicates that the shape function 0 – _ ; _ 0 ; used to weight the integrals and the shape function nþ1 ¼ n þ tð1 Þ n nþ1 ¼ 0 for interpolation are distinct. iteration step i ¼ 0 ½83 244 Numerical Methods for Mantle Convection

2. Solving: Here we will present governing equations and numer- Y ical methods for solving these equations (see also _ i e e _ i e e i M nþ1 ¼ ðfnþ1 – m nþ1 – ðk þ c Þnþ1Þ½84 Chapters 7.02 and 7.10). e 3. Corrector: 7.05.5.1.1 Governing equations Governing equations for thermochemical convection i þ1 ¼ i þ t_ i ; _ i þ 1 ¼ _ i þ _ i nþ1 nþ1 nþ1 nþ1 nþ1 nþ1 include a transport equation that describes the move- ½85 ment of compositions, in addition to the conservation 4. If needed, set iteration step i ¼ i þ 1 and go back laws of the mass, momentum, and energy (i.e., eqns step 2. [1]–[3]). Suppose that C describes the compositional field, the transport equation is We make four remarks for this algorithm. First, this method is second-order accurate if ¼ 0.5 (Hughes, qC þ ui C;i ¼ 0 ½86 2000). Second,Q typically two iterations are sufficient. qt Third, in [84] represents the operation of assem- This transport equation is similar to the energy bling elemental matrix into global matrix, and M is equation [3] except that chemical diffusion and the lumped mass matrix which essentially makes this source terms are ignored, which is justified given scheme an explicit scheme. Fourth, time increment that chemical diffusion is likely extremely small for t needs to satisfy Courant time-stepping constraints the length- and timescales that we consider in mantle to make the scheme stable (Hughes, 2000). convection. For a two-component system such as the crust–mantle system or depleted-primordial mantle system, C can be either 0 or 1, representing either 7.05.5 Incorporating More Realistic component. If the fluids of different compositions Physics have intrinsically different density, then the momen- tum eqn [2] needs to be modified to take into account In Section 7.05.2, we presented the governing equa- the compositional effects on the buoyancy tions for thermal convection in a homogeneous þ RaðT – CÞ ¼ 0 ½87 incompressible fluid with a Newtonian (linear) ij ; j iz rheology and the Boussinesq approximation. where is the buoyancy number (van Keken et al., However, the ’s mantle is likely much more 1997; Tackley and King, 2003) and is defined as complicated with heterogeneous composition and ¼ =ðTÞ½88 non-Newtonian rheology (see Chapter 7.02). In addi- tion, non-Boussinesq effects such as solid–solid phase where is the density difference between the two transitions may play an important role in affecting the compositions, and T are the reference density dynamics of the mantle. In this section, we will discuss and temperature, and is the reference coefficient of the methods that help incorporate these more realistic thermal expansion. physics in studies of mantle convection. We will focus A special class of thermochemical convection pro- on modeling thermochemical convection, solid-state blems examine how the mantle compositional phase transitions, and non-Newtonian rheology. heterogeneity is stirred by mantle convection (e.g., Gurnis and Davies, 1986; Christensen, 1989; Kellogg, 1992; van Keken and Zhong, 1999). For these studies 7.05.5.1 Thermochemical Convection on the mixing of the mantle, we may assume that the Thermal convection for a compositionally heteroge- fluids with different compositions have identical den- neous mantle has gained a lot of interest in recent sity with ¼ 0. years (Lenardic and Kaula, 1993; Tackley, 1998a; Davaille, 1999; Kellogg et al., 1999; Chapter 7.10), 7.05.5.1.2 Solution approaches with focus on the roles of mantle compositional Solving the conservation equations of the mass, anomalies and crustal structure in mantle dynamics. momentum, and energy for thermochemical convec- This is also called thermochemical convection. tion is identical to what was introduced in Section Different from purely thermal convection for which 7.05.3 for purely thermal convection. The additional the fluid has the same composition, thermochemical compositional buoyancy term in the momentum convection involves fluids with different compositions. equation [87] does not present any new difficulties Numerical Methods for Mantle Convection 245

numerically, provided that the composition C is element/grid cell that includes type 1 particles N1 given. The new challenge is to solve the transport and type 2 particles N2 is given as equation [86] effectively. A number of techniques have been developed or Ce ¼ N2=ðN1 þ N2Þ½90 adapted in solving the transport equation in thermo- In the ratio method, C can never be greater than 1. chemical convection studies. They include a field Tackley and King (2003) found that the ratio method method with a filter (e.g., Hansen and Yuen, 1988; is particularly effective in modeling thermochemical Lenardic and Kaula, 1993), a marker chain method convection in which the two components occupy (Christensen and Yuen, 1984; van Keken et al., 1997; similar amount of volumes. Zhong and Hager, 2003), and a particle method (e.g., We now discuss briefly procedures to update the Weinberg and Schmeling, 1992; Tackley, 1998a; positions of particles. One commonly used method is Tackley and King, 2003; Gerya and Yuen, 2003). As a high-order Runge–Kutta method (e.g., van Keken reviewed by van Keken et al. (1997), while these et al., 1997). Here we present a predictor–corrector techniques work to some extent, they also have scheme for updating the particle positions (e.g., their limitations, particularly in treating entrainment Zhong and Hager, 2003). Suppose that at time t ¼ t0, and numerical diffusion of composition C. We will flow velocity is u0 and compositional field is C0 that is briefly discuss each of these methods with more i defined by a set of particles with coordinates, x0, for emphasis on the particle method. particle i. The algorithm for solving composition at In the particle method, the transport equation for the next time step t ¼ t0 þ dt ¼ t1, C1, can be C (i.e., eqn [86]) is not solved directly. Composition C summarized as follows: at a given time is represented by a set of particles. This representation requires a mapping from the 1. Using a forward Euler scheme, predict the new i i distribution of particles to compositional field C position for each particle i with x1p ¼ x0 þ u0dt which is often represented on a numerical mesh. and mapping the particles to compositional field With the mapping, to update C, all that is needed is C1p at t ¼ t1. to update the position of each particle to obtain an 2. Using the predicted C1p, solve the Stokes equation updated distribution of particles. This effectively for new velocity u1p. solves the transport equation for C. 3. Using a modified Euler scheme with second-order Two different particle methods have been used to accuracy, compute the position for each particle i i i map distribution of particles to C: absolute and ratio with x1 ¼ x0 þ 0:5ðu0 þ u1pÞdt and composi- methods (Tackley and King, 2003). In the absolute tional field C1 at t ¼ t1. method, particles are only used to represent one type The marker chain method is similar to the particle of composition (e.g., for dense component or with method in a number of ways. In the marker chain C ¼ 1). The population density of particles can be method, composition C is defined by an interfacial mapped to C. For example, C for an element/grid cell boundary that separates two different components. with volume e and particles Ne can be given as The interfacial boundary is a line for 2-D problems or a surface for 3-D. Using the flow velocity, one C ¼ AN = ½89 e e e tracks the evolution of the interfacial boundary and where the constant A is the reciprocal of initial den- hence composition C. Often the interfacial boundary sity of particles for composition C ¼ 1 (i.e., total is represented by particles or markers. Therefore, number of particles divided by the volume of com- updating the interfacial boundary is essentially the position C ¼ 1). Clearly, the absence of particles in an same as updating the particles in the particle method. element/grid cell represents C ¼ 0. A physically Composition C on a numerical grid which is desired unrealistic situation with C > 1 may arise due to for solving the momentum and energy equations [87] statistical fluctuations in particle distribution or par- and [3] can be obtained by projection. As van Keken ticle settling. Therefore, for this method to work et al. (1997) indicated, the marker chain method is effectively, a large number of particles are required rather effective for compositional anomalies with (Tackley and King, 2003). relatively simple structure and geometry in 2-D. In the ratio method, two different types of parti- The field method is probably the most straightfor- cles are used to represent the compositional field C, ward. By setting diffusivity to be zero, we can employ type 1 for C ¼ 0 and type 2 for C ¼ 1. C for an the same solver for the energy equation (e.g., in 246 Numerical Methods for Mantle Convection

Section 7.05.4.3) to solve the transport equation for C. kth phase transition. After normalizing pressure by 0 However, this often introduces numerical artifacts gD and Clapeyron slope by 0gD=T, the nondi- including numerical oscillations and numerical diffu- mensional ‘excess pressure’ can be written as sion. Lenardic and Kaula (1993) introduced a filter scheme that removes the numerical oscillations while k ¼ 1 – dk – z – kðT – TkÞ½94 conserving the total mass of compositional field. where k; dk, and Tk are the nondimensional Clapeyron slope, reference-phase transition depth, 7.05.5.2 Solid-State Phase Transition and reference-phase transition temperature for the kth phase transition, respectively. The dimensionless Solid-state phase transitions are important phenom- phase-change function is then given as ena in the mantle. Major phase transitions include olivine–spinel transition at 410 km depth and spinel- 1 to-perovskite and magnesium-to-wustite transitions ¼ 1 þ tanh k ½95 k 2 d at 670 km depth, that are associated with significant changes in mantle density and speeds. where d is dimensionless phase transition width Recently, it was proposed that the D99 discontinuity which measures the depth segment over which the near the core–mantle boundary is also caused by a phase change occurs. It should be pointed out that the phase transition from perovskite to post-perovskite effect of phase transition on buoyancy force can also (Murakami, et al., 2004). These phase transitions may be modeled with ‘effective’ coefficient of thermal affect the dynamics of mantle convection in two expansion (Christensen and Yuen, 1985). ways: (1) on the energetics due to latent heating The latent heating effect, along with viscous heat- associated with phase transitions, (2) on the buoyancy ing and adiabatic heating, can be included in the due to undulations at phase boundary caused by energy equation (also see eqn [3]) as (Christensen lateral variations in mantle temperature that affects and Yuen, 1985) the pressure at which phase transitions occur X (Richter, 1973; Schubert et al., 1975; Christensen Ra d qT 1 þ 2 k k D ðT þ T Þ þ v ? rT and Yuen, 1985; Tackley et al., 1993; Zhong and k i s q k Ra d k t Gurnis, 1994). In this section, following Richter X Ra d (1973) and Christensen and Yuen (1985), we present þ 1 þ k k ðT þ T ÞD v k Ra d s i z a method to model phase transitions. k k The undulations of a phase boundary represent 2 Di qvi ¼rT þ ij þ ½96 additional buoyancy force that affects the momentum Ra qxj equation. For phase transition k with density change where Ts, vz,andij are the surface temperature, k, the phase boundary undulations for phase transi- vertical velocity, and deviatoric stress, respectively; tion k with density change k can be described by a k is phase-change index; Di is the dissipation number dimensionless phase-change function k that varies and is defined as from 0 to 1 where regions with k of0and1represent the two phases separated by this phase-change bound- gD ary. The momentum equation can be written as Di ¼ ½97 Cp ij ; j þðRaT – RakkÞiz ¼ 0 ½91 k where and Cp are the coefficient of thermal expan- where phase-change Rayleigh number Rak is sion and specific heat (see also Chapter 7.02). Christensen and Yuen (1985) called the effects of gD3 Ra ¼ k ½92 latent heating, viscous heating, and adiabatic heating k 0 as non-Boussinesq effects and termed this formulation The phase-change function k is defined via as extended-Boussinesq formulation. They suggest ‘excess pressure’ that these effects are all of similar order, proportional to D , and should be considered simultaneously. ¼ P – P – T ½93 i k 0 k The modified momentum and energy equations where k and P0 are the Clapeyron slope and phase- [91] and [96] can be solved with the same algorithms change pressure at zero-degree temperature for the such as the Uzawa and SUPG for mantle convection Numerical Methods for Mantle Convection 247 problems with extended-Boussinesq approximations converge very slowly or may diverge. Often some in 3-D (e.g., Zhong, 2006; Kameyama and Yuen, 2006). forms of damping may help improve convergence significantly (e.g., King and Hager, 1990).

7.05.5.3 Non-Newtonian Rheology Laboratory studies suggest that the deformation of 7.05.6 Concluding Remarks and olivine, the main component in the , Future Prospects follows a power-law rheology (e.g., Karato and Wu, 1993): In this chapter, we have discussed four basic numerical methods for solving mantle convection problems: FE, "_ ¼ A n ½98 FD, FV, and spectral methods. We have focused our where "_ is the strain rate, is the deviatoric stress, the efforts on FE method, mainly because of its growing pre-exponent constant A represents other effects such popularity in mantle convection studies over the past as grain size and water content, and the exponent n is decade, partially prompted by the easily accessed FE 3. The nonlinearity in the rheology arises from codes from Conman to Citcom. To this end, the dis- n 6¼ 1(see also Chapter 7.02). cussions on FE method should help readers The effects of non-Newtonian rheology on man- understand the inner working of these two FE codes. tle convection were first investigated by Parmentier However, our discussions on FD, FV, and spectral et al. (1976) and Christensen (1984). More recent methods are rather brief and are meant to give efforts have been focused on how non-Newtonian readers a solid basis for understanding the rudiments rheology including viscoplastic rheology may lead of these methods and the references with which to to dynamic generation of (King and delve deeper into the subjects. These three methods Hager, 1990; King et al., 2002; Bercovici, 1995; Moresi have all been widely used in studies of mantle con- and Solomatov, 1998, Zhong et al., 1998; Tackley, vection and will most likely remain so for years to 1998b; Trompert and Hansen, 1998). come. It is our view that each of these methods has its Solutions of nonlinear problems in general require advantages and disadvantages and readers need to find an iterative approach. The power-law rheolgy may the one that is most suited to their research. This be written as an expression for effective viscosity chapter is by no means exhaustive or extremely advanced in character. We did not cover many poten- ij ¼ – Pij þ 2eff "_ij ½99 tially interesting and powerful numerical techniques, such as spectral elements (e.g., Komatitsch and 1 ð1 – nÞ=n eff ¼ ="_ ¼ "_ ½100 Tromp, 1999), wavelets (Daubeschies et al., 1985), A level-set method (Osher and Fedkiew, 2003), and where "_ is the second invariant of strain-rate tensor adaptive grid techniques (Berger and Oliger 1984; Bruegmann and Tichy, 2004), and interested readers 1 1=2 "_ ¼ "_ "_ ½101 can read these references to learn more. 2 ij ij Rapid advancement in computing power has made It is clear that the effective viscosity depends on 3-D modeling of mantle convection practical, strain rate which in turn depends on flow velocity. although 2-D modeling still plays an essential role. Therefore, a general strategy for this problem is While a variety of solvers with either iterative or (1) starting with some guessed effective viscosity, direct solution method are available to 2-D models, solve the Stokes flow problem for flow velocities; (2) 3-D modeling requires iterative solution techniques update the effective viscosity with the newly deter- due to both computer memory and computational mined strain rate, and solve the Stokes flow again; requirements. A powerful iterative solution approach (3) keep this iterative process until flow velocities are is the multigrid method that can be used in either FE convergent. or FV method. Such methods are already implemen- Implementation of this iterative scheme is ted in a number of mantle convection codes including straightforward. The convergence for this iterative STAG3D (Tackley, 1994), Citcom/CitcomS (Moresi process depends on the exponent n. For regular and Solomatov, 1995; Zhong et al., 2000), and Terra power-law rheology with n 3, convergence is (Baumgardner, 1985). We spent some effort in discuss- usually not a problem. However, for large n (e.g., in ing the multigrid method, and more on this topic can case of viscoplastic rheology), the iteration may be found in Brandt (1977), Trottenberg et al. (2001), 248 Numerical Methods for Mantle Convection and Yavneh (2006). The multigrid idea is powerful in models. 2-D compressible mantle that one can generalize this to structures other than convection models with simple thermodynamics grids, for instance multiscale or multilevel techniques have been formulated (Jarvis and McKenzie, 1980; (Trottenberg et al., 2001). Ita and King, 1998). We anticipate more develop- Three-dimensional modeling should almost cer- ments in the near future. (4) Multiscale physics is tainly make a good use of parallel computing with an important feature of mantle convection. Earth’s widely available parallel computers from PC-clusters mantle convection is of very long wavelengths, for to super-parallel computers. Parallel computing example, 10 000 km for the Pacific Plate. However, technology poses certain limitations on numerical mantle convection is also fundamentally controlled techniques as well. For example, spectral methods, by thin thermal boundary layers that lead to thin while having some important advantages over other upwelling plumes (100 km) and slabs grid-based numerical methods, are much more diffi- due to high Rayleigh number in mantle convection. cult to implement efficiently in parallel computing. Plate boundary processes and entrainment in ther- This may severely limit its use to tackle next-gen- mochemical convection also occur over possibly eration computing problems. Fortunately, many even smaller length scale. Furthermore, material other numerical methods including FE, FD, and FV properties are also affected by near-microscopic methods are very efficient for parallel computing. properties such as grain size, which has not been Many of the codes mentioned earlier for mantle well incorporated in mantle convection studies. convection studies including STAG3D, Citcom/ Most existing numerical methods in mantle convec- CitcomS, Terra, and those in Harder and Hansen tion work for largely uniform grids. New methods (2005) and Kageyama and Sato (2004) are fully par- that work with dynamic adaptive mesh refinement allelized and can be used on different parallel are needed. Finally, it is necessary for all these new computers. These codes can be scaled up to possibly methods and algorithms to work efficiently in 2-D/ thousands of processors with a great potential yet to 3-D on parallel computers. be explored in helping understand high-resolution, Efficient postprocessing and analyses of modeling high-Rayleigh-number mantle convection. results are also increasingly becoming an important However, there remain many challenges in issue. Mantle convection, along with many other numerical modeling of mantle convection, both in disciplines in the geosciences, now faces an exponen- developing more robust numerical algorithms and tial increase in the amount of numerical data in analyzing model results. At least four areas need generated in large-scale high-resolution 3-D convec- better numerical algorithms. (1) Thermochemical tion. It is currently commonplace to have a few convection becomes increasingly important in terabytes of data for a single project and this poses answering a variety of geodynamic questions. We significant challenges to conventional ways of discussed cursorily a Lagrangian technique of data analyses, post-processing, and visualization. advection of tracers in solving thermochemical con- Visualization is already a severe problem even in vection. However, it is clear that most existing the era of terascale computing. Considering our techniques do not work well for entrainment in ther- future quest to the petascale computing, this problem mochemical convection, as demonstrated by van would be greatly exacerbated in the coming decade. Keken et al. (1997) and Tackley and King (2003). It is essential to employ modern visualization tools to The increasing computing power will help solve confront this challenge. Erlebacher et al. (2001) dis- this problem by providing significantly high resolu- cussed these issues and their solutions, and similar tion, but better algorithms are certainly needed. arguments about these problems and their solutions (2) The is characterized by highly non- can be found in a report on high-performance com- linear rheology including complex shear-localizing puting in geophysics by Cohen (2005). Several feedback mechanisms and history-dependent rheol- potentially useful methodologies such as 2-D/3-D ogy and plastic deformation (Gurnis et al., 2000; feature extraction, segmentation methods, and flow Bercovici, 2003). Convergence deteriorates rapidly topology (see Hansen and Johnson, 2005) can help when nonlinearity increases. More robust algorithms geophysicists understand better the physical struc- are needed for solving mantle convection with highly ture of time-dependent convection with coherent nonlinear rheology. 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