Local characteristic decomposition based central-upwind scheme
Introduction
This paper focuses on numerical solutions of hyperbolic systems of conservation laws, which in the two-dimensional (2-D) case, read as where x and y are spatial variables, t is the time, is a vector of unknowns, and 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 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].
Section snippets
1-D central-upwind scheme: a brief overview
In this section, we consider the 1-D hyperbolic system of conservation laws and describe the second-order semi-discrete CU scheme from [18]. To this end, we assume that the computational domain is covered with the uniform cells of size Δx centered at and denote by cell averages of over the corresponding intervals , that is, We also assume that at certain time , the cell average values are available and from here on we
1-D LCD-based central-upwind scheme
In this section, we introduce a new 1-D LCD-based CU scheme, in which the amount of numerical dissipation is substantially reduced compared with the original CU scheme (2.2)–(2.3). To this end, we first rewrite the numerical flux (2.3) of the original CU scheme in the following form: where and is the numerical diffusion given by which, in
2-D LCD-based central-upwind scheme
In this section, we generalize the 1-D LCD-based CU scheme introduced in §3 for the 2-D hyperbolic system of conservation laws (1.1). We design the 2-D LCD-based CU scheme in a “dimension-by-dimension” manner, so that it reads as where Here, , , and and are the numerical diffusion terms defined by
Numerical examples
In this section, we apply the proposed LCD-based CU schemes, which will be referred to as the New CU schemes, to the 1-D and 2-D Euler equations of gas dynamics described in Appendices A and C, respectively. We conduct several numerical experiments and compare the performance of the New CU schemes with that of the corresponding 1-D and 2-D second-order CU schemes from [18] and [22], respectively, which will be referred to as the Old CU schemes.
In Examples 1–11, we take the specific heat ratio
CRediT authorship contribution statement
Alina Chertock: Conceptualization, Funding acquisition, Investigation, Methodology, Visualization, Writing – original draft, Writing – review & editing. Shaoshuai Chu: Conceptualization, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing – original draft, Writing – review & editing. Michael Herty: Conceptualization, Funding acquisition, Investigation, Methodology, Visualization, Writing – original draft, Writing – review & editing. Alexander Kurganov:
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
The work of A. Chertock was supported in part by NSF grants DMS-1818684 and DMS-2208438. The work of M. Herty was supported in part by the DFG (German Research Foundation) through 20021702/GRK2326, 333849990/IRTG-2379, HE5386/18-1, 19-2, 22-1, 23-1 and under Germany's Excellence Strategy EXC-2023 Internet of Production 390621612. The work of A. Kurganov was supported in part by NSFC grants 12171226 and 12111530004, and by the fund of the Guangdong Provincial Key Laboratory of Computational
References (46)
- et al.
Semi-discrete central-upwind Rankine-Hugoniot schemes for hyperbolic systems of conservation laws
J. Comput. Phys.
(2021) On the treatment of contact discontinuities using WENO schemes
J. Comput. Phys.
(2011)- et al.
New high-resolution semi-discrete central schemes for Hamilton-Jacobi equations
J. Comput. Phys.
(2000) A numerical study of the performance of alternative weighted ENO methods based on various numerical fluxes for conservation law
Appl. Math. Comput.
(2017)- et al.
Nonoscillatory central differencing for hyperbolic conservation laws
J. Comput. Phys.
(1990) - et al.
Characteristic finite-difference WENO scheme for multicomponent compressible fluid analysis: overestimated quasi-conservative formulation maintaining equilibriums of velocity, pressure, and temperature
J. Comput. Phys.
(2017) - et al.
Low shear diffusion central schemes for particle methods
J. Comput. Phys.
(2020) - et al.
On the construction, comparison, and local characteristic decomposition for high-order central WENO schemes
J. Comput. Phys.
(2002) - et al.
Resolution of high order WENO schemes for complicated flow structures
J. Comput. Phys.
(2003) - et al.
Efficient implementation of essentially non-oscillatory shock-capturing schemes
J. Comput. Phys.
(1988)
An improved fifth order alternative WENO-Z finite difference scheme for hyperbolic conservation laws
J. Comput. Phys.
Généralisation du schéma de Nessyahu-Tadmor pour une équation hyperbolique à deux dimensions d'espace
C. R. Acad. Sci. Paris Sér. I Math.
Generalized Riemann Problems in Computational Fluid Dynamics
A characteristic-wise alternative WENO-Z finite difference scheme for solving the compressible multicomponent non-reactive flows in the overestimated quasi-conservative form
J. Sci. Comput.
Numerical Analysis of Compressible Fluid Flows
Computing oscillatory solutions of the Euler system via -convergence
Math. Models Methods Appl. Sci.
On the computation of measure-valued solutions
Acta Numer.
Symmetric hyperbolic linear differential equations
Commun. Pure Appl. Math.
A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics
Mat. Sb.
Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations
Strong stability-preserving high-order time discretization methods
SIAM Rev.
Numerical Methods for Conservation Laws: From Analysis to Algorithms
Nonoscillatory central schemes for multidimensional hyperbolic conservation laws
SIAM J. Sci. Comput.
Cited by (6)
New Low-Dissipation Central-Upwind Schemes
2023, Journal of Scientific ComputingAdaptive High-Order A-WENO Schemes Based on a New Local Smoothness Indicator
2023, East Asian Journal on Applied MathematicsLocal Characteristic Decomposition Based Central-Upwind Scheme for Compressible Multifluids
2023, Springer Proceedings in Mathematics and Statistics