Optimized low-dissipation and low-dispersion schemes for compressible flows

Abstract Although interpolation-based finite difference methods have shown great success in the simulation of complex configurations, the schemes currently are still inappropriate in many circumstances due to their poor spectral properties. In this paper, a class of optimized low-dissipation and low-dispersion schemes is developed with the cell-centered finite difference method (Liao, J. Comput. Phys., 284:419–433, 2015). Firstly, the linear schemes with two free parameters which dominate the dispersion and dissipation properties independently are derived. Secondly, the two free parameters are optimized separately according to specific criteria to obtain the optimized linear schemes. Thirdly, for shock-capturing capability, the optimized non-linear schemes are acquired from the optimized linear schemes using the multi-stencil weighting technique. Meanwhile, a new series of weighting functions and smoothness indicators are derived following the idea of Hu (Hu, J. Comput. Phys., 229:8952–8965, 2010) for the optimized non-linear schemes. Finally, several benchmark cases are validated with the developed optimized linear and non-linear schemes. In flows with shocks, the optimized non-linear schemes can give more flow details than the non-linear reference schemes without noticeable robustness loss. In the low-speed problems, the optimized linear schemes perform much better than the corresponding upwind linear schemes. Additionally, satisfactory results are observed on the wavy grid, indicating the optimized schemes with cell-centered finite difference method are promising.

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

[2]  Taku Nonomura,et al.  Increasing Order of Accuracy of Weighted Compact Non-Linear Scheme , 2007 .

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

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

[5]  S. Orszag,et al.  Small-scale structure of the Taylor–Green vortex , 1983, Journal of Fluid Mechanics.

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

[7]  P. Lax.  Weak solutions of nonlinear hyperbolic equations and their numerical computation , 1954 .

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

[9]  Xiaogang Deng,et al.  A family of hybrid cell-edge and cell-node dissipative compact schemes satisfying geometric conservation law , 2015 .

[10]  Chi-Wang Shu,et al.  Development of nonlinear weighted compact schemes with increasingly higher order accuracy , 2008, J. Comput. Phys..

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

[12]  Fei Liao,et al.  Extending geometric conservation law to cell-centered finite difference methods on moving and deforming grids , 2015, J. Comput. Phys..

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

[14]  Sanjiva K. Lele,et al.  High-order localized dissipation weighted compact nonlinear scheme for shock- and interface-capturing in compressible flows , 2017, J. Comput. Phys..

[15]  Fei Liao,et al.  Extending geometric conservation law to cell-centered finite difference methods on stationary grids , 2015, J. Comput. Phys..

[16]  Chi-Wang Shu,et al.  A new class of central compact schemes with spectral-like resolution II: Hybrid weighted nonlinear schemes , 2015, J. Comput. Phys..

[17]  Xiaogang Deng,et al.  Developing high-order weighted compact nonlinear schemes , 2000 .

[18]  Taku Nonomura,et al.  Robust explicit formulation of weighted compact nonlinear scheme , 2013 .

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

[20]  Taku Nonomura,et al.  Freestream and vortex preservation properties of high-order WENO and WCNS on curvilinear grids , 2010 .

[21]  Yu-Xin Ren,et al.  A class of finite difference schemes with low dispersion and controllable dissipation for DNS of compressible turbulence , 2011, J. Comput. Phys..

[22]  Parviz Moin,et al.  Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves , 2010, J. Comput. Phys..

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

[24]  Nikolaus A. Adams,et al.  An adaptive central-upwind weighted essentially non-oscillatory scheme , 2010, J. Comput. Phys..

[25]  Taku Nonomura,et al.  Effects of difference scheme type in high-order weighted compact nonlinear schemes , 2009, J. Comput. Phys..

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

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

[28]  Miguel R. Visbal,et al.  On the use of higher-order finite-difference schemes on curvilinear and deforming meshes , 2002 .

[29]  Takuji Kurotaki,et al.  A new central compact finite difference formula for improving robustness in weighted compact nonlinear schemes , 2015 .

[30]  Zhenguo Yan,et al.  Further improvement of weighted compact nonlinear scheme using compact nonlinear interpolation , 2017 .

[31]  C. Tam,et al.  Dispersion-relation-preserving finite difference schemes for computational acoustics , 1993 .

[32]  Lei Luo,et al.  A sixth order hybrid finite difference scheme based on the minimized dispersion and controllable dissipation technique , 2014, J. Comput. Phys..

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