Solution of two‐dimensional Riemann problems for gas dynamics without Riemann problem solvers

We report here on our numerical study of the two-dimensional Riemann problem for the com- pressible Euler equations. Compared with the relatively simple 1-D congurations, the 2-D case consists of a plethora of geometric wave patterns which pose a computational challenge for high- resolution methods. The main feature in the present computations of these 2-D waves is the use of the Riemann-solvers-free central schemes presented in (11). This family of central schemes avoids the intricate and time-consuming computation of the eigensystem of the problem, and hence oers a considerably simpler alternative to upwind methods. The numerical results illustrate that despite their simplicity, the central schemes are able to recover with comparable high-resolution, the various features observed in the earlier, more expensive computations. AMS subject classication: Primary 65M10; Secondary 65M05

[1]  Stanley Osher,et al.  Nonoscillatory high order accurate self-similar maximum principle satisfying shock capturing schemes I , 1996 .

[2]  Doron Levy,et al.  A Third-Order Semi-Discrete Central Scheme for Conservation Laws and Convection-Diffusion Equations , 2000, math/0002133.

[3]  Alexander Kurganov,et al.  CENTRAL SCHEMES AND CONTACT DISCONTINUITIES , 2000 .

[4]  Gui-Qiang Chen,et al.  On the 2-D Riemann problem for the compressible Euler equationsI. Interaction of shocks and rarefaction waves , 1995 .

[5]  S. Osher,et al.  Regular ArticleUniformly High Order Accurate Essentially Non-oscillatory Schemes, III , 1997 .

[6]  Alexander Kurganov,et al.  Semidiscrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton-Jacobi Equations , 2001, SIAM J. Sci. Comput..

[7]  E. Tadmor,et al.  New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection—Diffusion Equations , 2000 .

[8]  Doron Levy,et al.  A Third-Order Semidiscrete Central Scheme for Conservation Laws and Convection-Diffusion Equations , 2000, SIAM J. Sci. Comput..

[9]  Eitan Tadmor,et al.  Nonoscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws , 1998, SIAM J. Sci. Comput..

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

[11]  Chi-Wang Shu Total-variation-diminishing time discretizations , 1988 .

[12]  Gui-Qiang Chen,et al.  On the 2-D Riemann problem for the compressible Euler equations II. Interaction of contact discontinuities , 2000 .

[13]  ShuChi-Wang,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes, II , 1989 .

[14]  Stanley Osher,et al.  Convex ENO High Order Multi-dimensional Schemes without Field by Field Decomposition or Staggered Grids , 1998 .

[15]  S. Osher,et al.  On the convergence of difference approximations to scalar conservation laws , 1988 .

[16]  Xu-Dong Liu,et al.  Solution of Two-Dimensional Riemann Problems of Gas Dynamics by Positive Schemes , 1998, SIAM J. Sci. Comput..

[17]  Carsten W. Schulz-Rinne,et al.  The Riemann Problem for Two-Dimensional Gas Dynamics , 1993 .

[18]  Gabriella Puppo,et al.  On the behavior of the total variation in CWENO methods for conservation laws , 2000 .

[19]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[20]  G. Tadmor,et al.  Non-oscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws , 1997 .

[21]  Z. Xin,et al.  The relaxation schemes for systems of conservation laws in arbitrary space dimensions , 1995 .

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

[23]  James P. Collins,et al.  Numerical Solution of the Riemann Problem for Two-Dimensional Gas Dynamics , 1993, SIAM J. Sci. Comput..

[24]  E. Tadmor,et al.  Non-oscillatory central differencing for hyperbolic conservation laws , 1990 .

[25]  Gabriella Puppo,et al.  A third order central WENO scheme for 2D conservation laws , 2000 .

[26]  C. Angelopoulos High resolution schemes for hyperbolic conservation laws , 1992 .

[27]  Alexander Kurganov,et al.  A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems , 2001, Numerische Mathematik.

[28]  E. Tadmor,et al.  Third order nonoscillatory central scheme for hyperbolic conservation laws , 1998 .

[29]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

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

[31]  P. Lax Weak solutions of nonlinear hyperbolic equations and their numerical computation , 1954 .

[32]  Gabriella Puppo,et al.  Compact Central WENO Schemes for Multidimensional Conservation Laws , 1999, SIAM J. Sci. Comput..

[33]  Centro internazionale matematico estivo. Session,et al.  Advanced Numerical Approximation of Nonlinear Hyperbolic Equations , 1998 .

[34]  Tong Zhang,et al.  Conjecture on the structure of solutions of the Riemann problem for two-dimensional gas dynamics systems , 1990 .

[35]  Eitan Tadmor,et al.  New High-Resolution Semi-discrete Central Schemes for Hamilton—Jacobi Equations , 2000 .