Approximate Riemann Solvers for Fluid Flow with Material Interfaces

The construction of Godunov-type schemes at material interfaces is considered. The main building block of the flux calculation in a Godunov-type scheme is the exact or the approximate solution of Riemann problems. At a material interface the flux function is discontinuous due to the change of the equation of state. A regularization of this problem is introduced and a Roe linearization of the regularized Riemann problem is given. From its solution appropriate numerical fluxes at material interfaces are deduced. These can be applied directly in the framework of Godunov-type schemes for the Euler equations in the Lagrangian frame of reference or on a moving grid. In Eulerian coordinates fixed in space it must be combined with an appropriate tracking of the material interfaces.

[1]  Claus-Dieter Munz,et al.  On Godunov-type schemes for Lagrangian gas dynamics , 1994 .

[2]  James M. Hyman,et al.  Numerical methods for tracking interfaces , 1984 .

[3]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[4]  G. Sod.  A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws , 1978 .

[5]  Phillip Colella,et al.  Efficient Solution Algorithms for the Riemann Problem for Real Gases , 1985 .

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

[7]  C. Munz,et al.  Simple Godunov-Type Schemes for Lagrangean Gas Dynamics , 1993 .

[8]  Claus-Dieter Munz,et al.  On the numerical dissipation of high resolution schemes for hyperbolic conservation laws , 1988 .

[9]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[10]  P. Lax.  Hyperbolic systems of conservation laws II , 1957 .

[11]  Randolph G. Smith.  The Riemann problem in gas dynamics , 1976 .

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

[13]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[14]  Ralph Menikoff,et al.  Errors When Shock Waves Interact Due to Numerical Shock Width , 1994, SIAM J. Sci. Comput..

[15]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

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

[17]  S. F. Davis,et al.  An interface tracking method for hyperbolic systems of conservation laws , 1992 .

[18]  John K. Dukowicz,et al.  Accurate conservative remapping (rezoning) for arbitrary Lagrangian-Eulerian computations , 1987 .

[19]  J. Marsden,et al.  A mathematical introduction to fluid mechanics , 1979 .

[20]  R. Menikoff,et al.  The Riemann problem for fluid flow of real materials , 1989 .

[21]  E. Halter,et al.  A fast solver for Riemann problems , 1985 .