A New Smoothness Indicator for the WENO Schemes and Its Effect on the Convergence to Steady State Solutions
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The convergence to steady state solutions of the Euler equations for the fifth-order weighted essentially non-oscillatory (WENO) finite difference scheme with the Lax–Friedrichs flux splitting [7, (1996) J. Comput. Phys. 126, 202–228.] is studied through systematic numerical tests. Numerical evidence indicates that this type of WENO scheme suffers from slight post-shock oscillations. Even though these oscillations are small in magnitude and do not affect the “essentially non-oscillatory” property of WENO schemes, they are indeed responsible for the numerical residue to hang at the truncation error level of the scheme instead of settling down to machine zero. We propose a new smoothness indicator for the WENO schemes in steady state calculations, which performs better near the steady shock region than the original smoothness indicator in [7, (1996) J. Comput. Phys. 126, 202–228.]. With our new smoothness indicator, the slight post-shock oscillations are either removed or significantly reduced and convergence is improved significantly. Numerical experiments show that the residue for the WENO scheme with this new smoothness indicator can converge to machine zero for one and two dimensional (2D) steady problems with strong shock waves when there are no shocks passing through the domain boundaries.
KeywordsWENO scheme steady state solution smoothness indicator
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- 5.Jameson A. (1993). Artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence in transonic and hypersonic flows. AIAA Paper 93–3359Google Scholar
- 15.Shu, C.-W. (1998). Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor (Editor: A. Quarteroni), Advanced Numerical Approximation of Nonlinear Hyperbolic Equations Lecture Notes in Mathematics, Vol. 1697. Springer, Berlin, pp. 325–432.Google Scholar
- 21.Zhang, S., Zhang, Y.-T., and Shu, C.-W. (2005). Multistage interaction of a shock wave and a strong vortex. Phys. Fluid 17, 116101.Google Scholar
- 22.Zhang Y.-T., Shi J., Shu C.-W., Zhou Y. (2003). Numerical viscosity and resolution of high-order weighted essentially nonoscillatory schemes for compressible flows with high Reynolds numbers. Phys. Rev. E 68, 046709Google Scholar