A hybrid numerical method and its application to inviscid compressible flow problems

Abstract An improved version of the artificially upstream flux vector scheme, is developed to efficiently compute inviscid compressible flow problems. This numerical scheme, named AUFSR (Tchuen et al. 2011), is obtained by hybridizing the AUFS scheme with Roe’s solver. This approach handles difficulties encountered by the AUFS scheme, in the case where the flux vector does not check the homogeneous property. The present scheme for multi-dimensional flows introduces a certain amount of numerical dissipation to shear waves, as Roe’s splitting. The AUFSR scheme is not only robust for shock-capturing, but also accurate for resolving shear layers. Numerical results for 1D Riemann problems and several 2D problems are investigated to show the capability of the method to accurately compute inviscid compressible flow when compared to AUFS, and Roe solvers.

[1]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[2]  B. Leer,et al.  Flux-vector splitting for the Euler equations , 1997 .

[3]  B. W. Skews,et al.  The perturbed region behind a diffracting shock wave , 1967, Journal of Fluid Mechanics.

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

[5]  Kazuyoshi Takayama,et al.  An artificially upstream flux vector splitting scheme for the Euler equations , 2003 .

[6]  Jean-Marc Moschetta,et al.  Shock wave numerical structure and the carbuncle phenomenon , 2005 .

[7]  J. Steger,et al.  Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods , 1981 .

[8]  Jean-Marc Moschetta,et al.  Robustness versus accuracy in shock-wave computations , 2000 .

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

[10]  P. Woodward,et al.  The numerical simulation of two-dimensional fluid flow with strong shocks , 1984 .

[11]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[12]  J. Quirk.  A Contribution to the Great Riemann Solver Debate , 1994 .

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

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

[15]  Stanley Osher,et al.  UNIFORMLY HIGH ORDER ACCURATE , .

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

[17]  G. Tchuen,et al.  Computation of non-equilibrium hypersonic flow with artificially upstream flux vector splitting (AUFS) schemes , 2008 .

[18]  Paul Woafo,et al.  Hybrid upwind splitting scheme by combining the approaches of Roe and AUFS for compressible flow problems , 2011 .

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

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

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

[22]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme III. Upstream-centered finite-difference schemes for ideal compressible flow , 1977 .

[23]  Dong Yan,et al.  Cures for numerical shock instability in HLLC solver , 2011 .