Local characteristic decomposition based central-upwind scheme

Highlights

• A new robust and accurate tool for numerically solving hyperbolic systems of conservation laws is provided.

• The proposed numerical methods are based on the local characteristic decomposition applied to central-upwind schemes.

• The new schemes have been implemented for the one- and two-dimensional Euler equations of gas dynamics.

• Numerical examples demonstrate the high accuracy and high resolution of the proposed schemes.

• The new central-upwind schemes outperform the original ones even though there is additional computational cost.

Abstract

We propose novel less diffusive schemes for conservative one- and two-dimensional hyperbolic systems of nonlinear partial differential equations (PDEs). The main challenges in the development of accurate and robust numerical methods for the studied systems come from the complicated wave structures, such as shocks, rarefactions and contact discontinuities, arising even for smooth initial conditions. In order to reduce the diffusion in the original central-upwind schemes, we use a local characteristic decomposition procedure to develop a new class of central-upwind schemes. We apply the developed schemes to the one- and two-dimensional Euler equations of gas dynamics to illustrate the performance on a variety of examples. The obtained numerical results clearly demonstrate that the proposed new schemes outperform the original central-upwind schemes.

Introduction

This paper focuses on numerical solutions of hyperbolic systems of conservation laws, which in the two-dimensional (2-D) case, read as

$${\mathit{U}}_t+\mathit{F}{(\mathit{U})}_x+\mathit{G}{(\mathit{U})}_y=\mathbf{0},$$

where x and y are spatial variables, t is the time, $\mathit{U}\in {\mathbb{R}}^d$ is a vector of unknowns, $\mathit{F}:{\mathbb{R}}^d\to {\mathbb{R}}^d$ and $\mathit{G}:{\mathbb{R}}^d\to {\mathbb{R}}^d$ are the x- and y-flux functions, respectively.

It is well-known that the solutions of (1.1) may develop complicated wave structures including shocks, rarefactions, and contact discontinuities even for infinitely smooth initial data, and thus developing highly accurate and robust shock-capturing numerical methods for solving (1.1) is a challenging task.

Since the pioneering works [7], [9], [24], a large number of various methods had been introduced; see, e.g., the monographs and review papers [2], [12], [16], [25], [39], [42] and references therein. Here, we focus on finite-volume (FV) methods, in which solutions are realized in terms of their cell averages and evolved in time according to the following algorithm. First, a piecewise polynomial interpolant is reconstructed out of the given cell averages, and then the evolution step is performed using the integral form of (1.1). To this end, a proper set of time-space control volumes has to be selected. Depending on this selection, one may distinguish between two basic classes of finite-volume methods: upwind and central schemes. In upwind schemes, the spatial part of the control volumes coincides with the FV cells and therefore, one needs to (approximately) solve (generalized) Riemann problems at every cell interface; see, e.g., [2], [9], [42] and references therein. This helps the upwind schemes to achieve very high resolution. At the same time, it might be quite complicated and even impossible to solve the (generalized) Riemann problems for general systems of conservation laws. Central schemes offer an attractive simple alternative to the upwind ones. In staggered central schemes, first proposed for one-dimensional (1-D) systems in [32] and then extended to higher order [26], [31] and multiple number of dimensions [1], [13], [27], the control volumes use the staggered spatial parts so that the FV cell interfaces remain inside the control volumes. This allows to avoid even approximately solving any Riemann problems, which makes staggered central schemes easy to implement for a wide variety of hyperbolic systems. The major drawback of staggered central schemes is, however, their relatively large numerical dissipation as they basically average over the Riemann fans rather than resolving them.

In order to reduce the amount of excessive numerical dissipation present in central schemes, a class of central-upwind (CU) schemes have been proposed in [18], [21]. These schemes are based on nonuniform control volumes, whose spatial size taken to be proportional to the local speeds of propagation. This allows one to minimize the area over which the solution averaging occurs still without (approximately) solving any (generalized) Riemann problems. The upwind features of the CU schemes can be seen, for example, when they are applied to simpler systems. For instance, the CU scheme from [18] reduces to the upwind one when it is applied to a system whose Jacobian contains only positive (only negative) eigenvalues. Another advantage of the CU schemes is related to the fact that unlike the staggered central schemes, they admit a particularly simple semi-discrete form. This observation is the basis of the modifications of the CU schemes we propose in this paper.

Even though the CU schemes from [18], [21] are quite accurate, efficient and robust tools for a wide variety of hyperbolic systems, higher resolution of the numerical solutions can be achieved by further reducing numerical dissipation. This can be done in a number of different ways, for example: (i) by introducing a more accurate evolution procedure, which leads to a “built-in” anti-diffusion term [20]; (ii) by implementing a numerical dissipation switch to control the amount of numerical dissipation present in the CU schemes [17]; (iii) by obtaining more accurate estimates for the one-sided local speeds of propagation using the discrete Rankine-Hugoniot conditions [8].

Another way to control the amount of numerical dissipation present in the CU schemes or any other FV methods is by adjusting the nonlinear limiting mechanism used in the piecewise polynomial reconstruction. It is well-known that sharper reconstructions may lead to larger numerical oscillations and a way to reduce these oscillations is to reconstruct the characteristic variables rather than the conservative ones; see, e.g., [35]. This can be done using the local characteristic decomposition (LCD); see, e.g., [3], [14], [15], [30], [33], [35], [39], [44] and references therein.

In this paper, we modify the CU schemes from [18] by applying LCD to the numerical diffusion part of the schemes. The obtained new LCD-based CU schemes contain substantially smaller amount of numerical dissipation, which leads to a significantly improved resolution of the computed solution compared with the original CU schemes. As observed above, the key idea is that the new LCD-based CU scheme reduces to the upwind scheme when applied to 1-D linear hyperbolic systems

$${\mathit{U}}_t+A{\mathit{U}}_x=\mathbf{0},$$

where A is a constant matrix (disregarding the sign of the eigenvalues of A). This feature suggests that the new CU schemes have more built-in upwinding compared with their predecessors.

The paper is organized as follows. In §2, we briefly describe the 1-D second-order FV CU scheme from [18]. In §3, we introduce the proposed new 1-D LCD-based CU scheme and show that the developed 1-D scheme reduces to the upwind scheme when applied to the linear system (1.2). In §4, we construct the 2-D LCD-based CU scheme. Finally, in §5, we test the proposed schemes on a number of 1-D and 2-D numerical examples for the Euler equations of gas dynamics. We demonstrate high accuracy, efficiency, stability, and robustness of the new LCD-based CU schemes, which clearly outperform the second-order CU scheme from [18], [22].