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On a Practical Implementation of Particle Methods Applied Numerical Mathematics 56 (2006) 1418–1431 www.elsevier.com/locate/apnum On a practical implementation of particle methods Alina Chertock a,∗, Alexander Kurganov b a Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA b Mathematics Department, Tulane University, New Orleans, LA 70118, USA Available online 18 May 2006 Abstract This paper is devoted to a practical implementation of deterministic particle methods for solving transport equations with dis- continuous coefficients and/or initial data, and related problems. In such methods, the solution is sought in the form of a linear combination of the delta-functions, whose positions and coefficients represent locations and weights of the particles, respectively. The locations and weights of the particles are then evolved in time according to a system of ODEs, obtained from the weak formulation of the transport PDEs. The major theoretical difficulty in solving the resulting system of ODEs is the lack of smoothness of its right-hand side. While the existence of a generalized solution is guaranteed by the theory of Filippov, the uniqueness can only be obtained via a proper regularization. Another difficulty one may encounter is related to an interpretation of the computed solution, whose point values are to be recovered from its particle distribution. We demonstrate that some of known recovering procedures, suitable for smooth functions, may fail to produce reasonable results in the nonsmooth case, and discuss several successful strategies which may be useful in practice. Different approaches are illustrated in a number of numerical examples, including one- and two-dimensional transport equations and the reactive Euler equations of gas dynamics. © 2006 IMACS. Published by Elsevier B.V. All rights reserved. MSC: 65M25; 35F10; 34A36; 76V05; 35L65 Keywords: Linear transport equations with discontinuous coefficients; Deterministic particle method; Method of characteristics; Finite-volume methods; Reactive Euler equations of gas dynamics 1. Introduction We consider the linear transport equation: ϕt + divx(uϕ) = S(x,t,ϕ), (1) subject to the initial condition: ϕ(x,t)= ϕ0(x). (2) d d Here, u ∈ R and ϕ0 ∈ R are functions of x ∈ R and t (not necessarily smooth), and S is a (possibly singular point) source term. If the coefficient u and the data ϕ0 and S are sufficiently smooth, it is easy to prove that the * Corresponding author. E-mail addresses: [email protected] (A. Chertock), [email protected] (A. Kurganov). URLs: www.math.ncsu.edu/~acherto, www.math.tulane.edu/~kurganov. 0168-9274/$30.00 © 2006 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.apnum.2006.03.024 A. Chertock, A. Kurganov / Applied Numerical Mathematics 56 (2006) 1418–1431 1419 initial-value problem (1)–(2) has a unique classical solution. When the transport velocity is possibly discontinuous, various existence–uniqueness–stability theories have evolved with a different definition of a solution and different conditions on the transport velocity (see, e.g., [43,4]). The problem is even more complicated in the case of several space dimensions, where the main difficulty is that generalized flows in the sense of PDE, called transport flows, are not unique and the weak stability of the conservation equation is presently out of reach, see, for example, [11,40,6,5] and the references therein. We also refer the reader to [21], where several numerical schemes for one-dimensional (1-D) transport equations with discontinuous coefficients were studied. Particle methods have been used for a long time to give a numerical solution of purely convective problems, such as the incompressible Euler equation in fluid mechanics [23,32,33] or the Vlasov equation in plasma physics [19]. In these methods, the solution is sought as a linear combination of Dirac distributions: N = − p ϕN (x,t) αi(t)δ x xi (t) , (3) i=1 { p } { } whose positions, xi (t) , and coefficients, αi(t) , represent locations and weights of the particles, respectively. The solution is then found by following the time evolution of the locations and the weights of the particles according to a system of ODEs, obtained by considering a weak formulation of the problem and substituting (3) into (1)–(2). In order to recover a proper approximation of the solution ϕ(x,t) at some time t>0, one needs to regularize the particle solution ϕN (x,t), and hence the performance of the method depends on the quality of the regularization procedures, allowing the recovery of the approximate solution from its particle distribution. In the context of linear transport equations with sufficiently smooth coefficients and initial data, the particle methods have been analyzed in [41]. We also refer the reader to [12] for a general introduction of the particle methods and their numerical analysis and to [9] for the study of the convergence rate of these methods. When a particle method is applied to problems with nonsmooth data, the most challenging part of the overall algorithm is recovering point values of the computed solution from its particle distribution. A commonly used type of the reconstruction is the regularization of a particle solution performed by taking a convolution with a so-called cutoff function, which plays the role of a smooth approximation to a delta-function. This procedure works perfectly fine for smooth functions, but may break down when applied to nonsmooth (discontinuous) solutions: depending on the width of the convolution kernel, one may get either a smeared or an oscillatory approximation, as it is demonstrated in Sections 2 and 3. In this paper, we consider different algorithms for reconstruction of numerical solutions from particle distributions and, in particular, suggest a strategy, which, in most cases, seems to provide the best solution of the recovery problem. We first use a convolution with a low width cutoff function, which results in an oscillatory approximation of the point values, which are then filtered by the nonlinear filter proposed in [15] for the treatment of solution cell averages. Even though this strategy has worked reasonably well in all the examples (both one- and two-dimensional) we considered, no convergence proof is available. An alternative recovery approach we use in this paper is based on the concept of a dual solution. It was proposed in [7] in the context of the transport equation that describes propagation of passive pollutant in shallow water. Here, we extend it to treat the equation for the mass fraction of the burnt/unburnt gases in reactive flows. The resulting particle method is similar (and in some cases, is equivalent) to the method of characteristics, and provides one with a set of computed point values rather then a particle distribution of the solution. A similar situation occurs when advection of a scalar quantity in a given incompressible flow field is considered. We would also like to emphasize the advantage of using particle methods for solving the transport equation (1) with a stiff term S. Such equations arise in many different applications, but in this paper we are concerned with the 1-D Euler equations that model a time-dependent flow of an inviscid, compressible reactive gas: ρ ρu ρu + ρu2 + p = 0, (4) E t u(E + p) x where ρ,u,p, and E are the density, velocity, pressure, and the total energy, respectively. The transport equations for the mass fraction of the unburnt gas z: 1 (ρz) + (ρuz) =− ρzK(τ), (5) t x ε 1420 A. Chertock, A. Kurganov / Applied Numerical Mathematics 56 (2006) 1418–1431 is coupled with the system (4) through the equations of state (EOS): ρu2 p = (γ − 1) · E − − q ρz, ,γ,q= const, (6) 2 0 0 and the Arrhenius kinetics term [13]: − K(τ) = e τc/τ , (7) where τ := p/ρ is the temperature and τc is the ignition temperature. When 0 <ε 1, the reaction is fast and the time scale associated with the stiff reaction term is much smaller than that associated with the fluid advection. The main difficulty in solving such equations numerically is obtaining the correct propagation speed of the detonation waves in the case of underresolved computations, that is, when both temporal ( t) and spatial ( x) grid scales are much larger than ε. In such a case, one may want to use an operator splitting (fractional step) method, and to replace the solution of the corresponding reaction ODE: 1 (ρz) =− ρzK(τ), (8) t ε with the deterministic projection operator. However, as it was first observed in [10], such a deterministic projection method may lead to a spurious weak detonation wave that travels with an unphysical propagation speed of one grid cell per time step. This is a result of numerical dissipation present in shock-capturing methods. To overcome this difficulty, several method have been proposed (see, e.g., [1–3,24,25,27,38]). In Section 4, we demonstrate that applying low dissipative, mesh-free particle methods to this model results in an accurate computed solution even when a problem with a complicated wave interactions is considered. 2. Particle methods for transport equations with discontinuous coefficients We begin with a brief description of the particle method for the initial-value problem (1)–(2). As it is mentioned in Section 1, the solution is sought as a linear combination of Dirac distributions given by (3). Considering a weak formulation of the problem and substituting (3) into (1)–(2) results in the following system of ODEs for the locations, p xi (t), and the weights, αi(t), of the particles: ⎧ ⎨ dxp(t) i = p dt u(xi ,t), ⎩ (9) dαi (t) = dt βi(t), p where βi reflects the contribution of the source term S (see, e.g., [41,9]). The initial positions of the particles, xi (0), and their initial weights, αi(0), are chosen to provide a high-order approximation to the initial data (2), that is, to accurately approximate the integral equation N p = αi(0)ψ xi (0) ϕ0(x)ψ(x) dx = i 1 X 1 for all C0 test functions ψ.
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