Overcoming shock instability of the HLLE-type Riemann solvers

Abstract This paper concentrates on numerical shock instabilities of some approximate Riemann solvers which stem from the HLLE method, including HLLC-type and HLLEM-type flux functions. Carbuncle phenomena are illustrated for these numerical schemes, especially for those believed to be carbuncle-free, such as the HLLE scheme. A linear matrix stability analysis shows that there exists a threshold independent of shock strength to trigger instability for a given numerical scheme. In this instability mechanism, a specific cell interface in the numerical structure of shock layer is regarded as the source of instability if the density ratio between both side of this interface exceeds the threshold. The modification by adjusting nonlinear signal velocities in these numerical schemes can change solution behaviors in the shock layer, and reduce the degree of density transition on the specific interface. A very simple cure strategy for the HLLE-type Riemann solvers is presented based on the in-depth understanding of the carbuncle onset of viscous shock.

[1]  S. F. Davis.  Simplified second-order Godunov-type methods , 1988 .

[2]  Domenic D'Ambrosio,et al.  Numerical Instablilities in Upwind Methods: Analysis and Cures for the “Carbuncle” Phenomenon , 2001 .

[3]  A. Harten.  High Resolution Schemes for Hyperbolic Conservation Laws , 2017 .

[4]  Hua Li,et al.  On numerical instabilities of Godunov-type schemes for strong shocks , 2017, J. Comput. Phys..

[5]  Keiichi Kitamura,et al.  Very simple, carbuncle-free, boundary-layer-resolving, rotated-hybrid Riemann solvers , 2008, J. Comput. Phys..

[6]  Yu-Xin Ren,et al.  A robust shock-capturing scheme based on rotated Riemann solvers , 2003 .

[7]  Michael Dumbser,et al.  A matrix stability analysis of the carbuncle phenomenon , 2004 .

[8]  Guangwei Yuan,et al.  A Stability Analysis of Hybrid Schemes to Cure Shock Instability , 2014 .

[9]  James J. Quirk,et al.  A Contribution to the Great Riemann Solver Debate , 1994 .

[10]  Meng-Sing Liou,et al.  Mass Flux Schemes and Connection to Shock Instability , 2000 .

[11]  Jishan Hu,et al.  Projection Dynamics in Godunov-Type Schemes , 1998 .

[12]  Bram van Leer,et al.  Use of a rotated Riemann solver for the two-dimensional Euler equations , 1993 .

[13]  R. Hillier,et al.  Computation of shock wave diffraction at a ninety degrees convex edge , 1991 .

[14]  S. Imlay,et al.  Blunt-body flow simulations , 1988 .

[15]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[16]  Guangwei Yuan,et al.  A robust HLLC-type Riemann solver for strong shock , 2016, J. Comput. Phys..

[17]  Keiichi Kitamura,et al.  Evaluation of Euler Fluxes for Hypersonic Flow Computations , 2009 .

[18]  Meng-Sing Liou,et al.  An Accurate and Robust Flux Splitting Scheme for Shock and Contact Discontinuities , 1997, SIAM J. Sci. Comput..

[19]  Bernd Einfeld.  On Godunov-type methods for gas dynamics , 1988 .

[20]  Timothy J. Barth.  Some notes on shock resolving flux functions. Part 1: Stationary characteristics , 1989 .

[21]  Richard Sanders,et al.  Multidimensional Dissipation for Upwind Schemes , 1998 .

[22]  P. Roe,et al.  On Godunov-type methods near low densities , 1991 .

[23]  Derek M. Causon,et al.  On the Choice of Wavespeeds for the HLLC Riemann Solver , 1997, SIAM J. Sci. Comput..

[24]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .

[25]  E. Toro,et al.  Restoration of the contact surface in the HLL-Riemann solver , 1994 .

[26]  S. F. Davis,et al.  A rotationally biased upwind difference scheme for the euler equations , 1984 .

[27]  Jang-Hyuk Kwon,et al.  On the dissipation mechanism of Godunov-type schemes , 2003 .