New non-oscillatory central schemes on unstructured triangulations for hyperbolic systems of conservation laws

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Abstract

We discuss an extension of the Jiang–Tadmor and Kurganov–Tadmor fully-discrete non-oscillatory central schemes for hyperbolic systems of conservation laws to unstructured triangular meshes. In doing so, we propose a new, “genuinely multidimensional,” non-oscillatory reconstruction—the minimum-angle plane reconstruction (MAPR). The MAPR is based on the selection of an interpolation stencil yielding a linear reconstruction with minimal angle with respect to the horizontal. This means that the MAPR does not bias the solution by using a coordinate direction-by-direction approach to the reconstruction, which is highly desirable when unstructured meshes consisting of elements with (almost) arbitrary geometry are used. To show the “black-box solver” capabilities of the proposed schemes, numerical results are presented for a number of hyperbolic systems of conservation laws (in two spatial dimensions) with convex and non-convex flux functions. In particular, it is shown that, even though the MAPR is neither designed with the goal of obtaining a scheme that satisfies a maximum principle in mind nor is total-variation diminishing (TVD), it provides a robust non-oscillatory reconstruction that captures composite waves accurately.

Introduction

Non-oscillatory central schemes are a class of Godunov-type (i.e., shock-capturing, finite volume) numerical methods for solving hyperbolic systems of conservation laws (e.g., the Euler equations of gas dynamics). Throughout the last decade, central (Godunov-type) schemes have gained popularity due to their simplicity and efficiency. In particular, the latter do not require the solution of a Riemann problem or a characteristic decomposition to compute the intercell flux.

The first second-order accurate non-oscillatory central Godunov-type scheme was introduced by Nessyahu and Tadmor (NT) [1], whose work generalized the first-order accurate staggered Lax–Friedrichs scheme using a non-oscillatory piecewise-linear reconstruction in the spirit of van Leer’s MUSCL [2]. The NT scheme, which is one of the simplest and most versatile Godunov-type numerical methods, has recently been put on solid theoretical ground by the proof of the fully-nonlinear scheme’s convergence to the unique entropy solution of the problem in the case of strictly convex nonlinear scalar conservation laws [3]. In addition, over the last decade, the NT scheme has inspired a significant amount of research on the topic of non-oscillatory central schemes. Some of the recent work on central schemes includes, but is not limited to, semi-discrete formulations, less dissipative central-upwind schemes, extensions to multiple spatial dimensions and non-Cartesian meshes (see, e.g., [4], [5], [6], [7], [8], [9], [10], [11], [12], [13] and the references therein).

In this paper, we present an extension of the second-order accurate (in space and time) two-dimensional (2D) central scheme of Jiang and Tadmor (JT) [5], [13], which is the 2D version of the Nessyahu–Tadmor scheme, to unstructured triangulations. To be precise, however, we propose a Jiang–Tadmor-like scheme because the staggered control volumes are “fixed,” i.e., they are determined entirely from the non-staggered ones. However, unlike the schemes in [1], [5], in the extensions proposed herein we complete each time step with a reconstruction on the staggered mesh followed by a projection onto the original triangulation. The latter idea was first introduced by Jiang et al. [6] to “unstagger” the NT and JT schemes. The main advantage of this approach is that it makes the implementation of boundary conditions considerably simpler, and it does not require the staggered staggered mesh to coincide with the original one—a fact that is always true for Cartesian tensor-product grids but generally not true for unstructured meshes. With hindsight, we formulated the scheme in this manner to allow for the seamless incorporation of recent developments in central schemes, which we shall now discuss.

The JT scheme has been further refined by Kurganov and Tadmor (KT) [7] (the so-called modified central differencing scheme) by making use of the maximal local speeds of propagation in constructing the staggered mesh’s cells. In doing so, the scheme’s numerical diffusion becomes independent of the inverse of the time step size, which allows (in particular) a semi-discrete formulation of the method. In this paper, we also present an extension of the KT scheme to unstructured triangulations, which might be preferable (over the JT-type scheme) for certain problems due to its reduced numerical diffusion.

In light of the work of Kurganov and Petrova [11], which extended the “state-of-the-art” semi-discrete central-upwind schemes [8] to triangular meshes, we must provide further motivation for the present work. Our goal is to build a very simple fully-discrete central scheme on (truly) unstructured triangulations. This is unlike the central-upwind scheme in [11] because, even though the semi-discrete limit results in a “nice” closed-form expression for the system of ordinary differential equations (ODEs) governing the evolution of the cell averages, on a truly unstructured mesh the system of ODEs is different for each cell due to the latter’s dependance on local mesh-connectivity information. This presents serious problems for the implementation of adaptive mesh refinement. Moreover, for purely hyperbolic equations, it was reported in [7, p. 270] that the fully discrete central scheme’s performance is comparable to that of the semi-discrete one when the staggered mesh is built via the modified-central-differencing approach, which we show how to do for a triangulation.

Furthermore, in the context of the JT- and KT-type fully-discrete central schemes we discuss in this paper, we propose a novel “genuinely-multidimensional” reconstruction, which has the significant advantage (over those available in the literature) of being simple to formulate and implement. Therefore, our work differs fundamentally from that of Arminjon et al. [4], who proposed an extension of the Nessyahu–Tadmor scheme to 2D unstructured triangulations, because of our novel (and simpler) non-oscillatory reconstruction and staggered meshes that fit naturally into the hierarchy of central schemes [7], [8], [11]. In addition, an important point of the present work, supporting those in the literature, is that unstructured, adaptively-refined meshes can significantly improve the performance of a Godunov-type scheme by keeping discontinuities sharp with a minimal number of mesh elements.

An important goal of this paper, which will be accomplished in future work, is to set the floor for predictor–corrector-type algorithms that couple the (explicit, central) Godunov approach to conservation laws with the novel (implicit) L1-minimization finite element method of Guermond and Popov [14], [15]. In this vein, we also hope to show how many of the common tools (e.g., tessellation of arbitrary domains, error indicators and adaptive mesh refinement, etc.) of finite-element methods can be seamlessly incorporated into Godunov-type finite-volume schemes. This interconnection between the “classical” finite element building blocks and Godunov-type schemes would allow for the construction of algorithms and software for the computation of complex physical flows where, e.g., one must solve coupled systems of hyperbolic and elliptic equations. Such systems commonly arise in the modeling of fluid flow in oil reservoirs (see, e.g., the review article of Gerritsen and Durlofsky [16]), where computing the (nonlinear) advective contributions in the model proves to be most challenging. In this respect, as the recent work of Käser and Iske [17] shows, Godunov-type schemes (in conjunction with unstructured, adaptively refined meshes) have proven to be an effective tool for petroleum reservoir flow simulations.

Finally, this paper is organized as follows. In Section 1.1, the problem is stated and basic notation for the paper is set out. In Section 2, the numerical method is described, including the reconstruction, evolution and projection steps. Finally, in Section 3, several numerical examples, including scalar equations (with both convex and non-convex fluxes), two-phase reservoir flow and the Euler system of gas dynamics, are presented and discussed.

We consider the following initial-value problem for a 2D hyperbolic system of conservation laws:tq+xf(q)+yg(q)=0,(x,y,t)Ω×(0,T],q(x,y,t=0)=q0(x,y),(x,y)Ω,where ΩR2 is the interior of a polygonal domain, whose boundary we denote by Ω. In addition, let T be a conforming triangulation (see, e.g., [18, p. 56]) of Ω¯:=ΩΩ, i.e., a finite collection of, say, N subsets τ of Ω¯, called elements, each of which is a non-degenerate triangle (usually satisfying a minimum-angle condition). We denote by |τ| the area of an element τT.

Furthermore, w stands for the approximation to q, the true solution to (1), on the triangulation T. Then, the constant w¯in represents the approximate average of the solution over the element τiT at time t=tn, i.e.,w¯inq¯in:=1|τi|τiq(x,y,tn)dA,where dA:=dxdy. Moreover, given a fixed time step Δt, we define tn:=nΔt.

Finally, throughout the text we represent points in Euclidean space by ordered pairs, e.g. (x,y). However, if an ordered pair is followed by a superscript, e.g. (νx,νy), it stands for a (column) vector. In addition, a dot between two vectors denotes the usual Euclidean inner product.

Section snippets

Reconstruction

Without loss of generality, we restrict ourselves to the case of a scalar conservation law for the rest of this section. In the case of a system, the procedure described herein is applied to each component of w (i.e., each equation) in the same manner.

Thus, to approximate (1), we begin each time step with a piecewise-constant solution of the formw¯n(x,y)=i=1Nw¯inχi(x,y),where χi is the characteristic function of the element τi, i.e.,χi(x,y)=1,if(x,y)τi;0,otherwise.Then, we construct a

Numerical results

In this section, we present numerical results for a number of standard test problems and some new ones we propose. With the exception of the results in Section 3.1, we use the JT-type scheme for all calculations due to its simplicity and speed. In particular, since we compute the solution of equations with non-convex fluxes, the evaluation of the maximal local speed of propagation (10) becomes non-trivial (see, e.g., [12]); this complicates the implementation of the KT-type scheme.

Conclusions and future work

In this paper, we introduced a new family of fully-discrete high-order central Godunov-type schemes on unstructured triangulations. Along the way, we proposed a novel non-oscillatory reconstruction, the MAPR, based on the adaptive selection of the “flattest” interpolating plane. The MAPR is simple, effective and easy to implement on the staggered mesh as well as on the original triangulation. In this respect, our scheme is an improvement over the one presented in [4], where a different

Acknowledgements

This research was supported, in part, by NSF Grant DMS-0510650. In addition, we would like to express our thanks to Veselin Dobrev for providing us with a copy of, and his assistance with, AggieFEM, which we used as the code base for the implementation of the scheme proposed in this paper. All unstructured meshes used were generated using the software package NETGEN 4.4, and all figures were created using Matlab R2007a. Finally, we are indebted to Peter Popov for his valuable advice on various

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    Present address: Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208-3125, USA.

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