High-order adapter schemes for cell-centered finite difference method

Abstract The present paper proposes that reconstruction scheme and interpolation scheme can be converted into each other through two series of adapter schemes, which include reconstruction-to-interpolation (RI) adapter schemes and interpolation-to-reconstruction (IR) adapter schemes. For the high-order spatial discretization of the compressible Navier-Stokes equations, the RI adapter schemes can be used to derive interpolation schemes for the interpolation-based cell-centered finite difference method from the available optimized reconstruction schemes. The main advantage of the interpolation-based cell-centered finite difference method is the capability to realize high-order discretization on curvilinear grids with both shock-capturing capability and satisfaction of the geometric conservation law. In the present paper, we first derive the IR adapter schemes by comparing the difference schemes with their strong conservative forms. We then develop the corresponding RI adapter schemes by inversing the IR adapter schemes. Thereafter, the applications to the one-dimensional linear wave equation and the one-dimensional inviscid Burgers' equation have been briefly discussed. Finally, to demonstrate the application to three-dimensional Navier-Stokes equations, three highly optimized nonlinear reconstruction schemes are adapted into the corresponding interpolation ones through RI adapter schemes, which include WENO-CU6, WGVC-WENO7 and OMP6 schemes. The new interpolation schemes from adapters are compared with their original reconstruction ones through several benchmark cases. No noticeable robustness loss or accuracy loss has been found in these cases, indicating the effectiveness of the adapter schemes. No obvious increase in time cost has been observed, indicating the efficiency of the adapter schemes.

[1]  Shigeru Obayashi,et al.  Numerical (error) issues on compressible multicomponent flows using a high-order differencing scheme: Weighted compact nonlinear scheme , 2012, J. Comput. Phys..

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

[3]  Taku Nonomura,et al.  Geometric interpretations and spatial symmetry property of metrics in the conservative form for high-order finite-difference schemes on moving and deforming grids , 2014, J. Comput. Phys..

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

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

[6]  Yan Jiang,et al.  Free-stream preserving finite difference schemes on curvilinear meshes , 2014 .

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

[8]  Xian Liang,et al.  Nonlinear spectral-like schemes for hybrid schemes , 2014 .

[9]  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..

[10]  P. Thomas,et al.  Geometric Conservation Law and Its Application to Flow Computations on Moving Grids , 1979 .

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

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

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

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

[15]  Taku Nonomura,et al.  Conservative metric evaluation for high-order finite difference schemes with the GCL identities on moving and deforming grids , 2013, J. Comput. Phys..

[16]  H. C. Yee,et al.  Extension of Efficient Low Dissipative High Order Schemes for 3-D Curvilinear Moving Grids , 2000 .

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

[18]  Todd A. Oliver.  A High-Order, Adaptive, Discontinuous Galerkin Finite Element Method for the Reynolds-Averaged Navier-Stokes Equations , 2008 .

[19]  H. T. Huynh,et al.  A Flux Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin Methods , 2007 .

[20]  Xinliang Li,et al.  Optimized sixth‐order monotonicity‐preserving scheme by nonlinear spectral analysis , 2013 .

[21]  T. Nonomura,et al.  Application of central differencing and low‐dissipation weights in a weighted compact nonlinear scheme , 2017 .

[22]  Xiaogang Deng,et al.  New High-Order Hybrid Cell-Edge and Cell-Node Weighted Compact Nonlinear Schemes , 2011 .

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

[24]  Jeffrey A. Housman,et al.  A comparison of higher-order finite-difference shock capturing schemes , 2015 .

[25]  Xiaogang Deng,et al.  Weighted Compact High-Order Nonlinear Schemes for the Euler Equations , 1997 .

[26]  Huayong Liu,et al.  Further studies on Geometric Conservation Law and applications to high-order finite difference schemes with stationary grids , 2013, J. Comput. Phys..

[27]  Taku Nonomura,et al.  A new technique for freestream preservation of finite-difference WENO on curvilinear grid , 2015 .

[28]  Sergio Pirozzoli,et al.  On the spectral properties of shock-capturing schemes , 2006, J. Comput. Phys..

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

[30]  Eleuterio F. Toro,et al.  Finite-volume WENO schemes for three-dimensional conservation laws , 2004 .