An adaptive characteristic-wise reconstruction WENO-Z scheme for gas dynamic euler equations

Abstract Due to its excellent shock-capturing capability and high resolution, the WENO scheme family has been widely used in varieties of compressive flow simulation. However, for problems containing strong shocks and contact discontinuities, such as the Lax shock tube problem, the WENO scheme still produces numerical oscillations. To avoid such numerical oscillations, the characteristic-wise construction method should be applied. Compared to component-wise reconstruction, characteristic-wise reconstruction leads to much higher computational cost and thus is not suitable for large scale simulation such as direct numerical simulation of turbulence. In this paper, an adaptive characteristic-wise reconstruction WENO-Z scheme, i.e. the AdaWENO-Z scheme, is proposed to improve the computational efficiency of the characteristic-wise reconstruction method. By defining shared smoothness functions, shared smoothness indicators as well as shared WENO weights are firstly introduced to reduce the computational cost of the component-wise reconstruction procedure and to define a global switch function capable of detecting discontinuity. According to the given switch function, the new scheme performs characteristic-wise reconstruction near discontinuities and switches to component-wise reconstruction for smooth regions. Several one dimensional and two dimensional numerical tests are performed to validate and evaluate the AdaWENO-Z scheme. Numerical results show that AdaWENO-Z maintains essentially non-oscillatory flow field near discontinuities as with the characteristic-wise reconstruction method. Besieds, compared to the component-wise reconstruction method, AdaWENO-Z is about 20% to 40% faster which indicates its excellent efficiency.

[1]  G. A. Gerolymos,et al.  Very-high-order weno schemes , 2009, J. Comput. Phys..

[2]  Nail K. Yamaleev,et al.  Third-order Energy Stable WENO scheme , 2008, J. Comput. Phys..

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

[4]  Li Li,et al.  Preventing numerical oscillations in the flux-split based finite difference method for compressible flows with discontinuities , 2015, J. Comput. Phys..

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

[6]  Friedemann Kemm.  On the Proper Setup of the Double Mach Reflection as a Test Case for the Resolution of Gas Dynamics Codes , 2014 .

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

[8]  Jun Peng,et al.  A novel weighting switch function for uniformly high‐order hybrid shock‐capturing schemes , 2017 .

[9]  Nikolaus A. Adams,et al.  An efficient low-dissipation hybrid weighted essentially non-oscillatory scheme , 2012, J. Comput. Phys..

[10]  G. Russo,et al.  Central WENO schemes for hyperbolic systems of conservation laws , 1999 .

[11]  V. Gregory Weirs,et al.  A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence , 2006, J. Comput. Phys..

[12]  V. Guinot.  Approximate Riemann Solvers , 2010 .

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

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

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

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

[17]  Neil D. Sandham,et al.  Low-Dissipative High-Order Shock-Capturing Methods Using Characteristic-Based Filters , 1999 .

[18]  Jun Peng,et al.  Improvement of weighted compact scheme with multi-step strategy for supersonic compressible flow , 2015 .

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

[20]  Nail K. Yamaleev,et al.  A systematic methodology for constructing high-order energy stable WENO schemes , 2009, J. Comput. Phys..

[21]  Sergio Pirozzoli,et al.  Direct numerical simulation and analysis of a spatially evolving supersonic turbulent boundary layer at M=2.25 , 2004 .

[22]  Nikolaus A. Adams,et al.  A family of high-order targeted ENO schemes for compressible-fluid simulations , 2016, J. Comput. Phys..

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

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

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

[26]  S. Osher,et al.  Uniformly high order accuracy essentially non-oscillatory schemes III , 1987 .

[27]  Gabriella Puppo,et al.  CWENO: Uniformly accurate reconstructions for balance laws , 2016, Math. Comput..

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

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

[30]  Yiqing Shen,et al.  Multistep weighted essentially non-oscillatory scheme , 2014 .

[31]  Gabriella Puppo,et al.  Adaptive Application of Characteristic Projection for Central Schemes , 2003 .

[32]  Gecheng Zha,et al.  Improvement of weighted essentially non-oscillatory schemes near discontinuities , 2009 .

[33]  Gabriella Puppo,et al.  Cool WENO schemes , 2017, Computers & Fluids.