On the spectral properties of shock-capturing schemes

In this short note we analyze the performance of nonlinear, shock-capturing schemes in wavenumber space. For this purpose we propose a new representation for the approximate dispersion relation which accounts to leading order for nonlinear effects. Several examples are presented, which confirm that the present theory yields an improved qualitative representation of the true solution behavior compared to conventional representations. The theory can provide useful guidance for the choice of the most cost-effective schemes for specific applications, and may constitute a basis for the development of optimized ones.

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

[2]  Bernardo Cockburn,et al.  Nonlinearly stable compact schemes for shock calculations , 1994 .

[3]  C. Hirsch,et al.  Numerical Computation of Internal and External Flows. By C. HIRSCH. Wiley. Vol. 1, Fundamentals of Numerical Discretization. 1988. 515 pp. £60. Vol. 2, Computational Methods for Inviscid and Viscous Flows. 1990, 691 pp. £65. , 1991, Journal of Fluid Mechanics.

[4]  Sukumar Chakravarthy,et al.  High Resolution Schemes and the Entropy Condition , 1984 .

[5]  S. Lele.  Compact finite difference schemes with spectral-like resolution , 1992 .

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

[7]  T. Colonius,et al.  Computational aeroacoustics: progress on nonlinear problems of sound generation , 2004 .

[8]  J. Bowles,et al.  Fourier Analysis of Numerical Approximations of Hyperbolic Equations , 1987 .

[9]  T. Cheng,et al.  Numerical simulations of underexpanded supersonic jet and free shear layer using WENO schemes , 2005 .

[10]  Sergio Pirozzoli,et al.  Conservative Hybrid Compact-WENO Schemes for Shock-Turbulence Interaction , 2002 .

[11]  Charles Hirsch,et al.  Numerical computation of internal & external flows: fundamentals of numerical discretization , 1988 .

[12]  Yong-Tao Zhang,et al.  Resolution of high order WENO schemes for complicated flow structures , 2003 .

[13]  V. Gregory Weirs,et al.  Optimization of weighted ENO schemes for DNS of compressible turbulence , 1997 .

[14]  Chi-Wang Shu,et al.  Monotonicity Preserving Weighted Essentially Non-oscillatory Schemes with Increasingly High Order of Accuracy , 2000 .

[15]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[16]  Sergio Pirozzoli,et al.  Performance analysis and optimization of finite-difference schemes for wave propagation problems , 2007, J. Comput. Phys..