On the dissipation mechanism of upwind-schemes in the low Mach number regime: A comparison between Roe and HLL

It is well known that standard upwind schemes for the Euler equations face a number of problems in the low Mach number regime: stiffness, cancellation and accuracy problems. A new aspect of the accuracy problem, presented in this paper, is the dependence on the type of flux solver: while the accuracy of the HLL scheme massively decreases for Ma->0 on a given triangular mesh, the Roe scheme remains accurate, i.e. flows of arbitrarily small Mach numbers can - at least in principle - be simulated on a fixed triangular mesh. We give an asymptotic analysis of this phenomenon and present a number of numerical results.

[1]  R. J. R. Williams,et al.  An improved reconstruction method for compressible flows with low Mach number features , 2008, J. Comput. Phys..

[2]  M. Hafez,et al.  Frontiers of Computational Fluid Dynamics 2002 , 2001 .

[3]  D. Kröner.  Numerical Schemes for Conservation Laws , 1997 .

[4]  Georg Bader,et al.  The influence of cell geometry on the accuracy of upwind schemes in the low mach number regime , 2009, J. Comput. Phys..

[5]  G. Volpe.  Performance of compressible flow codes at low Mach numbers , 1993 .

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

[7]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[8]  C. L. Merkle,et al.  The application of preconditioning in viscous flows , 1993 .

[9]  H. Guillard,et al.  On the behaviour of upwind schemes in the low Mach number limit , 1999 .

[10]  Dimitris Drikakis,et al.  On entropy generation and dissipation of kinetic energy in high-resolution shock-capturing schemes , 2008, J. Comput. Phys..

[11]  Ralf Hartmann,et al.  A discontinuous Galerkin method for inviscid low Mach number flows , 2009, J. Comput. Phys..

[12]  Jörn Sesterhenn,et al.  On the Cancellation Problem in Calculating Compressible Low Mach Number Flows , 1999 .

[13]  Miloslav Feistauer,et al.  On a robust discontinuous Galerkin technique for the solution of compressible flow , 2007, J. Comput. Phys..

[14]  E. Turkel,et al.  Preconditioned methods for solving the incompressible low speed compressible equations , 1987 .

[15]  F. Rieper.  On the behaviour of numerical schemes in the low Mach number regime , 2008 .

[16]  Hervé Guillard.  On the behavior of upwind schemes in the low Mach number limit. IV: P0 approximation on triangular and tetrahedral cells , 2009 .

[17]  D A Caughey,et al.  Frontiers of computational fluid dynamics 1994 , 1994 .

[18]  Philip L. Roe,et al.  Characteristic time-stepping or local preconditioning of the Euler equations , 1991 .

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

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