A new mapped WENO scheme using order-preserving mapping

Existing mapped WENO schemes can hardly prevent spurious oscillations while preserving high resolutions at long output times. We reveal in this paper the essential reason for such phenomena. It is actually caused by that the mapping function in these schemes can not preserve the order of the nonlinear weights of the stencils. The nonlinear weights may be increased for non-smooth stencils and be decreased for smooth stencils. It is then indicated to require the set of mapping functions to be Order-Preserving in mapped WENO schemes. Therefore, we propose a new mapped WENO scheme with a set of mapping functions to be order-preserving which exhibits a remarkable advantage over the mapped WENO schemes in references. For long output time simulations, the new scheme has the capacity to attain high resolutions and avoid spurious oscillations near discontinuities meanwhile.

[1]  Rong Wang,et al.  An improved mapped weighted essentially non-oscillatory scheme , 2014, Appl. Math. Comput..

[2]  C. Schulz-Rinne,et al.  Classification of the Riemann problem for two-dimensional gas dynamics , 1991 .

[3]  Ruo Li,et al.  A modified adaptive improved mapped WENO method , 2021, Communications in Computational Physics.

[4]  A. Harten.  ENO schemes with subcell resolution , 1989 .

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

[6]  A. Chatterjee,et al.  Shock wave deformation in shock-vortex interactions , 1999 .

[7]  Thomas de Quincey.  [C] , 2000, The Works of Thomas De Quincey, Vol. 1: Writings, 1799–1820.

[8]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[9]  Yuxin Ren,et al.  A characteristic-wise hybrid compact-WENO scheme for solving hyperbolic conservation laws , 2003 .

[10]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[11]  Wai-Sun Don,et al.  Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes , 2013, J. Comput. Phys..

[12]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

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

[14]  G. Naga Raju,et al.  A modified fifth-order WENO scheme for hyperbolic conservation laws , 2016, Comput. Math. Appl..

[15]  M. D. Salas,et al.  A numerical study of two-dimensional shock vortex interaction , 1981 .

[16]  J. M. Powers,et al.  Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points , 2005 .

[17]  Chao Yang,et al.  A new smoothness indicator for improving the weighted essentially non-oscillatory scheme , 2014, J. Comput. Phys..

[18]  Pengxin Liu,et al.  Piecewise Polynomial Mapping Method and Corresponding WENO Scheme with Improved Resolution , 2015 .

[19]  Rong Wang,et al.  A modified fifth-order WENOZ method for hyperbolic conservation laws , 2016, J. Comput. Appl. Math..

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

[21]  Wai-Sun Don,et al.  An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws , 2008, J. Comput. Phys..

[22]  Ruo Li,et al.  An efficient mapped WENO scheme using approximate constant mapping , 2021, ArXiv.

[23]  T. H. New,et al.  Adaptive mapping for high order WENO methods , 2019, J. Comput. Phys..

[24]  Rong Wang,et al.  A New Mapped Weighted Essentially Non-oscillatory Scheme , 2012, J. Sci. Comput..

[25]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..

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

[27]  Wai-Sun Don,et al.  High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws , 2011, J. Comput. Phys..

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

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

[30]  Sergio Pirozzoli,et al.  Numerical Methods for High-Speed Flows , 2011 .

[31]  F. ARÀNDIGA,et al.  Analysis of WENO Schemes for Full and Global Accuracy , 2011, SIAM J. Numer. Anal..

[32]  Mengping Zhang,et al.  On the Order of Accuracy and Numerical Performance of Two Classes of Finite Volume WENO Schemes , 2011 .

[33]  S. Osher,et al.  Some results on uniformly high-order accurate essentially nonoscillatory schemes , 1986 .