A new formulation for two-wave Riemann solver accurate at contact interfaces

Abstract

This study proposes a new formulation for Harten, Lax and van Leer (HLL) type Riemann solver which is capable of solving contact discontinuities accurately but with robustness for strong shock. It is well known that the original HLL, which has incomplete wave structures, is too dissipative to capture contact discontinuities accurately. On the other side, contact-capturing approximate Riemann solvers such as Harten, Lax and van Leer with Contact (HLLC) usually suffer from spurious solutions, also called carbuncle phenomenon, for strong shock. In this work a new accurate and robust HLL-type formulation, so-called HLL-BVD (HLL Riemann solver with BVD) is proposed by modifying the original HLL with BVD (boundary variation diminshing) algorithm. Instead of explicitly recovering the complete wave structures like the way of HLLC, the proposed method restores the missing contact with a jump-like function. The capability of solving contact discontinuities is further improved by minimizing the inherent dissipation term in HLL. Without modifying the original incomplete wave structures of HLL, the robustness for strong shock has been reserved. Thus the proposed method is free from shock instability problem. The accuracy and robustness of the new method are demonstrated through solving several one-and two-dimensional tests. Results indicate that the new formulation based on two-wave HLL-type Riemann solver is not only capable of capturing contact waves more accurately than the original HLL or HLLC but, most importantly, is free form carbuncle instability for strong shock.

An exact Riemann solver for one‐dimensional multimaterial elastic‐plastic flows with Mie‐Grüneisen equation of state without vacuum

“…The final time is t = 85 s. Results of density and velocity shown in Figure 4 are compared the exact solution and numerical solution in Reference 16. This comparison verifies the correctness of our numerical scheme.…”

“…Even recently, Riemann problems still attract much attention, especially in the following three aspects. The first is to reduce numerical dissipation and to approve the robustness of schemes; 16,17 the second is to develop two‐dimensional Riemann solvers; 18‐20 and the third is to extend the existin approximate solvers to other systems, such as the magnetohydrodynamic (MHD) 18,21 and elastic‐plastic flows 22‐25 . In the study of Riemann problems, the exact Riemann solver has also played a very important role as it not only can give a guide and reference in constructing approximate Riemann solvers but also can be used to determine the convergence and stability of numerical schemes.…”

Numerical simulation and entropy dissipative cure of the carbuncle instability for the shallow water circular hydraulic jump

“…To do so we first use a combination of Rusanov's and Roe's solvers, switching between them based on a local measure of the entropy residual. Previous works have also proposed blending Riemann solvers with different amounts of dissipation in the context of the Euler equations (see [30,43,20,31,9,35]), the shallow water equations (see [2,21]), and even the Navier-Stokes equations (see [30,31]). Our approach differs from those just cited in that it is based on the local entropy; for a related approach in the context of the Euler equations, see [18,19].…”

Numerical simulation and entropy dissipative cure of the carbuncle instability for the shallow water circular hydraulic jump

“…To do so we first use a combination of Rusanov's and Roe's solvers, switching between them based on a local measure of the entropy residual. Previous works have also proposed blending Riemann solvers with different amounts of dissipation in the context of the Euler equations (see References 36‐41), the shallow water equations (see References 11 and 12), and even the Navier–Stokes equations (see References 36 and 39). Our approach differs from those just cited in that it is based on the local entropy; for a related approach in the context of the Euler equations, see References 6 and 16.…”

“…We model only the flow on the upstream side of the cylinder, since our interest is in the resolution of the bow shock. We take the domain [0, 40] × [0,100] with a cylinder of radius 20 centered at (40,50). Reflecting boundary conditions are imposed at the surface of the cylinder, along with outflow conditions along the rest of the right edge of the domain.…”