A Review of Finite-Element Methods for Solving the Shallow-Water Equations

A review of ﬁnite-element methods for solving the shallow-water equations

I.M. Navon Supercomputer Computations Research Institute and Department of Mathematics Florida State University, Tallahassee, FL 32306–4052

ABSTRACT: The shallow-water equations have been extensively used for a wide variety of coastal phenomena, such as tide-currents, pollutant- dispersion storm-surges, tsunami-wave propagation, etc., to mention but a few. In meteorology the shallow-water equations also known as the primitive barotropic equations have been extensively used to test new numerical solutions for numerical weather prediction models as they posses the same mixture of fast gravity waves and slow Rossby waves as the more complex three dimensional baroclinic primitive equations. In the present survey we will review the application of ﬁnite-element methods for solving the shallow-water equations. Various issues such as variable resolution, integral invariant conservation, etc. will be addressed.

the use of ﬁnite-element codes on vector su- percomputers. 1. INTRODUCTION

The solution of the shallow-water equations 2. SHALLOW-WATER EQUATIONS FOR is of considerable importance for a variety COASTAL MODELLING of problems of coastal and environmental engineering such as periodic (tidal) flow, transient wave phenomena (tsunami or It is commonly known that current flow in shock waves), transient pollutant transport, estuaries and coastal seas can be described seiches in ports, etc. by the shallow-water equations. Assuming We will review the different finite- vertical density gradients and fluid accelera- element techniques used for coastal phenom- tions are negligible the shallow-water equa- ena and briefly discuss them in Section 2. tions can be derived by integrating over the water-depth and assuming hydrostatic pres- In meteorology use of the finite-element sure. method was initiated by Wang et al., (1972) and the method has since then evolved substantially and is now considered a tool The equations are of preference by a sizable number of re- searchers seeking to solve the two dimensional shallow-water equations. The use of the finite-element method for solving the shallow-water equations in meteorology will ∂ξ + (HV) = 0 be discussed in Section 3. ∂t ∇ · In Section 4 we will briefly address (conservation of mass) (1) computational issues related to the use of ∂(HV) the finite-element method for solving the + (HVV) + τHV shallow-water equations, such as conserva- ∂t ∇ · + fk HV + gH ξ A = 0 (2) tion of integral invariants use of different × ∇ − types of elements, variable resolution and (conservation of momentum) Where automatic mesh generation was given by Praagman (1986). ξ is the surface elevation over mean sea level A conservative finite-element model of − h is bathymetry the shallow-water equation with linear tri- − angular elements using a two-step econom- H =h + τ is the total depth of flow ical algorithm was proposed by Peraire, V vertically averaged flow velocity − Zienkiewicz and Morgan (1986), similar τ nonlinear bottom friction (Chezy’s formula) to one proposed by Navon and Riphagen − f Coriolis parameter (1979) for a compact fourth-order conser- − vative finite-difference approximation of the g acceleration of gravity − shallow water equations. The method is eas- A atmospheric wind forcing ily amenable for vectorization. − k unit vector in vertical direction Apart from the basic work of Fix (1975), − little attention has been paid by coastal and ocean modelers to conservation of Early finite-element tidal models suf- integral invariants by finite-element models fered from severe spurious oscillations. The of the shallow-water equations. A selective early applications of these were controlled lumping finite-element method for shallow- by large bottom friction coefficients (see water flow has been extensively tested by Brebbia and Partridge, (1976)) or addition Kawahara et al., (1982). Platzman (1981) of damping to the model via time-stepping has considered some response characteristics schemes (Kawahara et al., (1978)). of finite-element tidal models with the view Gray and Lynch (1979) used semi- to partially eliminate small scale errors due implicit procedures to partially eliminate to the spatial discretization. spurious modes. Lynch and Gray (1979) presented a method called the wave equation approach 3. SHALLOW WATER EQUATIONS FOR which is in general insensitive to short METEOROLOGICAL FLOWS wavelength oscillations primarily due to good phase speed response. In meteorology the first application of the It is by now accepted that the wave finite-element method to the shallow-water equation approach is capable of noise sup- equations is due to Wang et al., (1972). pression in finite-element models without They solved the 1–D gravity-wave equations the need for artificial or unrealistic damping. u + uu + ghx =0 t x (3) Using different choices of basis functions ht + uhx + hux =0 for elevation and velocity (mixed interpolation) has been suggested by Hood and Tay- where u is the velocity of the fluid in the x- lor (1974) to eliminate 2∆x oscillations in direction, h is the depth of the fluid, and an attempt to imitate the use of staggered g is the acceleration of gravity. Cubic Her- grids in finite-difference approximations. mite functions were used on a uniform grid with a Crank-Nicolson time discretization Kinnmarck and Gray (1984) found method. Cullen (1974) used linear equilat- that use of a two time-level time-difference eral triangles and a leap-frog time scheme to approximation, symmetrical but for the non- solve the shallow-water equations written in linear convective terms can also eliminate the form: spurious 2∆t oscillations, in the velocity ut + uux + vuy + φx fv = 0 solutions, where the wave equation scheme − is used. vt + uvx + vvy + φy + fu = 0 (4) Triangular and quadratic Lagrangian φt + (uφ)x + (vφ)y = 0 isoparametric finite-elements are generally used, (Kinnmarck (1985)). Tidal and storm where u and v are the velocity compounds surge computations using triangular ele- in the x and y directions, φ = gh is the ments and quadratic interpolation were car- geopotential, f is the Coriolis parameter, ried out by Dalsecco et al., (1986) with vari- and h is the depth of the fluid, in a periodic able resolution. A harbour resonance prob- channel on a β-plane where lem for irregularly shaped harbours using f = f0 + βy . (5) Cullen (1974) and Hinsman (1975) used invariant conservation properties of different linear equilateral triangles to solve the finite-element schemes for the shallow-water shallow-water equations on the sphere for equations was conducted by Steppeler, the Rossby-Haurwitz waves. Staniforth and Navon, and Lee (1988). Determination of Mitchell (1977) used a two dimensional Cha- finite time “blow-up”, critical dissipativity peau basis function to solve the shallow- required to maintain nonlinear stability water equations on a polar stereographic and long-term integrations of the finite- projection. They used vorticity-divergence element shallow-water equations models formulation of the shallow-water equations. were researched. A variable resolution integration was per- formed by Staniforth and Mitchell (1978). Navon (1979) used an extrapolated 4. COMPUTATIONAL ASPECTS Crank-Nicolson scheme with linear triangular elements to solve the shallow-water While converge and accuracy estimates for equations on a β-plane. A selective lump- Galerkin finite-element methods applied to ing technique was implemented. Williams hyperbolic partial differential equations have (1981) has shown that finite-element formu- been extensively studied (see Dupont (1973) lation of the shallow-water equations using and Thomee and Wendroff (1974)), specific vorticity and divergence as predictive vari- evaluation of different issues for the finite- ables on an unstaggered grid does not suffer element methods which solve the shallow- from the same problems as unstaggered for- water equations were addressed by Cullen mulations in terms of velocity components (1976) and Navon (1977). (i.e., primitive forms of the equations). He A Fourier analysis for evaluating the ac- also concluded that if one uses velocity com- curacy of finite-element methods for the lin- ponents formulation, one should use them earized shallow-water equations, extended on a staggered grid. to include group velocity was presented by Williams and Zienkiewicz (1981) exam- Foreman (1982). His approach was based ined a new formulation of the shallow- water on similar work by Schoenstadt (1980) and wave equations using different basis func- Vichnevetsky and Peiffer (1975). In his re- tions for the velocity and height fields. They search, accuracy was the only consideration tested a staggered 1–D version of the lin- in determining a good method–an approach earized shallow-water equations. which is good for 1-D considerations. His Based on a proposal by Cullen and conclusions point out that the most accurate Morton (1980), Navon (1983) proposed a methods for wave amplitude, phase velocity, Numerov-Galerkin highly accurate finite- and group velocity may not coincide. element approach to the nonlinear advection He found that for a Galerkin finite- operator of the shallow-water equations. A element method with piecewise linear basis lucid review on the formulation of efficient functions the most accurate and stable finite-element codes for flows in regular do- two step method was the Crank-Nicolson mains was provided by Staniforth (1987). method. In a second study Foreman (1984) Neta et al., (1985) studied the linearized compared the accuracy and computational shallow-water equations with bilinear rect- cost of three finite-element methods for angular elements for a flow with variable solving the linearized, two-dimensional bottom topography. shalow-water equations. Neta and Williams (1986) studied var- He concluded that a finite-element ious finite-element formulations of the ad- scheme due to Thaker (1978) had some advection equation and found that isoceles vantage. For triangular elements his anal- triangles and rectangles with bilinear ba- ysis indicates that equilateral triangles are sis functions have better stability and phase the better choice–as they seem to produce speeds. Zienkiewicz and Heinrich (1979) and isotopic waves when the wave resolution is Zienkiewicz et al., (1986) proposed a finite- high. element-penalty approach. Steppeler (1987) Weare (1976) considered the computa- proposed an energy conserving finite-element tional cost of finite-element and finite differ- scheme for the shallow-water equations. ence methods for solving the shallow-water An in-depth research into the integral equations. He concluded that finite-element methods are computationally more expen- As far as time-discretization is con- sive due to the band algorithms used to cerned, explicit time-differencing schemes solve the sparse matrix equations resulting are not efficient for stiff sets of equations from this method of discretization. He sug- such as the shallow-water equations. gested iterative methods and lumping the Navon (1979, 1983) used a Crank- mass matrix as possible solutions. Nicolson method. Semi-implicit methods Staniforth (1987) shows that efficient were used by Staniforth and Daley (1979) solution algorithms are more easily derived for a baroclinic model and by Staniforth and for the rectangular element case than for Mitchell (1977) for the finite-element solu- the triangular element case using a tensor tion of the shallow-water equations. Semi- product method. In such a case he shows a implicit time-differencing constitutes a good way to vectorize the Gaussian elimination compromise for the shallow-water equations procedure. Staniforth (1987) makes also as they require fewer time-steps than ex- the remark that rectangular elements are plicit methods while their computational inherently more vectorizable because they cost per time-step is not significantly higher. are well-ordered in memory and easily As far as variable resolution is con- accessed. cerned, Older (1981) as well as Kelley Cullen and Morton (1980) and Navon and Williams (1976) have found that the (1983) applied a two-stage process (com- smoother and slower the change in resolu- pute the derivative, then the product) for tion, the better the forecast; and that one u∂v obtains improvement by concentrating high calculating the advection term ∂x , and resolution areas in the region of strongest have shown that the two-stage method has a gradients. 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