Abstract
With increasing engineering demands, there need high order accurate schemes embedded with precise physical information in order to capture delicate small scale structures and strong waves with correct “physics”. There are two families of high order methods: One is the method of line, relying on the Runge-Kutta (R-K) time-stepping. The building block is the Riemann solution labeled as the solution element “1”. Each step in R-K just has first order accuracy. In order to derive a fourth order accuracy scheme in time, one needs four stages labeled as “ 1⊙1⊙1⊙1=4”. The other is the one-stage Lax-Wendroff (LW) type method, which is more compact but is complicated to design numerical fluxes and hard to use when applied to highly nonlinear problems. In recent years, the pair of solution element and dynamics element, labeled as “2”, are taken as the building block. The direct adoption of the dynamics implies the inherent temporal-spatial coupling. With this type of building blocks, a family of two-stage fourth order accurate schemes, labeled as “ 2⊙2=4”, are designed for the computation of compressible fluid flows. The resulting schemes are compact, robust and efficient. This paper contributes to elucidate how and why high order accurate schemes should be so designed. To some extent, the “ 2⊙2=4” algorithm extracts the advantages of the method of line and one-stage LW method. As a core part, the pair “2” is expounded and LW solver is revisited. The generalized Riemann problem (GRP) solver, as the discontinuous and nonlinear version of LW flow solver, and the gas kinetic scheme (GKS) solver, the microscopic LW solver, are all reviewed. The compact Hermite-type data reconstruction and high order approximation of boundary conditions are proposed. Besides, the computational performance and prospective discussions are presented.
Highlights
1 Introduction In the simulation of compressible fluid flows or related problems, there are two families of commonly-used high order accurate numerical schemes: One is the family of methods of line, for which the fluid dynamical system is written in semi-discrete form and the Runge-Kutta (RK) temporal iteration is employed for the temporal discretization, such as RK-WENO [1], RK-discontinuous Galerkin (DG) [2] and their variants
Careful readers may observe the validity of (33) when the above two-stage algorithm applies to the current case, which is why we have to develop the Lax-Wendroff type flow solvers based on hyperbolic balance laws rather than the formal partial differential equations (Ben-Artzi, M, Li, J: On the consistency and convergence of finite volume approximations to hyperbolic balance laws, submitted)
8 Conclusion with prospective discussions It is natural to require the temporal-spatial coupling of a numerical method when simulating compressible fluid flows, for which the generalized Riemann problem (GRP) solver and the gas kinetic scheme (GKS) solver are reviewed briefly as the representatives of Lax-Wendroff type flow solvers
Summary
In the simulation of compressible fluid flows or related problems, there are two families of commonly-used high order accurate numerical schemes: One is the family of methods of line, for which the fluid dynamical system is written in semi-discrete form and the Runge-Kutta (RK) temporal iteration is employed for the temporal discretization, such as RK-WENO [1], RK-DG [2] and their variants. As strong waves (discontinuities) are present in flows (solutions), the CK procedure loses its physical and mathematical meanings, exhibiting the instability of the resulting schemes near discontinuities Careful inspection of these two families of methods motivates to combine the merits of both methods: The simplicity of multi-stage RK methods and the temporal-spatial coupling of LW type methods. In this paper we will elucidate the idea of this new family of schemes by interpreting the philosophy from ordinary differential equations (ODEs) to fluid dynamical systems, reviewing the well-used GRP and GKS solvers as the representatives of the Lax-Wendroff type solvers, building high order temporal-spatially coupled high order accurate schemes with favorable computational performance. The dynamical element is implicitly used in the construction of algorithm (3) This is why this method is of multi-stage two-derivative type with the pair (y, dy/dt) as the building block. This notation could be replaced by a better one
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