New optimized flux difference schemes for improving high-order weighted compact nonlinear scheme with applications

doi:10.1007/s10483-021-2712-8

摘要/Abstract

摘要： To improve the spectral characteristics of the high-order weighted compact nonlinear scheme (WCNS), optimized flux difference schemes are proposed. The disadvantages in previous optimization routines, i.e., reducing formal orders, or extending stencil widths, are avoided in the new optimized schemes by utilizing fluxes from both cell-edges and cell-nodes. Optimizations are implemented with Fourier analysis for linear schemes and the approximate dispersion relation (ADR) for nonlinear schemes. Classical difference schemes are restored near discontinuities to suppress numerical oscillations with use of a shock sensor based on smoothness indicators. The results of several benchmark numerical tests indicate that the new optimized difference schemes outperform the classical schemes, in terms of accuracy and resolution for smooth wave and vortex, especially for long-time simulations. Using optimized schemes increases the total CPU time by less than 4%.

关键词: optimization, flux difference, weighted compact nonlinear scheme (WCNS), resolution, spectral error

Abstract: To improve the spectral characteristics of the high-order weighted compact nonlinear scheme (WCNS), optimized flux difference schemes are proposed. The disadvantages in previous optimization routines, i.e., reducing formal orders, or extending stencil widths, are avoided in the new optimized schemes by utilizing fluxes from both cell-edges and cell-nodes. Optimizations are implemented with Fourier analysis for linear schemes and the approximate dispersion relation (ADR) for nonlinear schemes. Classical difference schemes are restored near discontinuities to suppress numerical oscillations with use of a shock sensor based on smoothness indicators. The results of several benchmark numerical tests indicate that the new optimized difference schemes outperform the classical schemes, in terms of accuracy and resolution for smooth wave and vortex, especially for long-time simulations. Using optimized schemes increases the total CPU time by less than 4%.

Key words: optimization, flux difference, weighted compact nonlinear scheme (WCNS), resolution, spectral error

中图分类号:

76M20
35Q35

Shichao ZHENG, Xiaogang DENG, Dongfang WANG. New optimized flux difference schemes for improving high-order weighted compact nonlinear scheme with applications[J]. Applied Mathematics and Mechanics (English Edition), 2021, 42(3): 405-424.

参考文献

[1] LIU, X. D., OSHER, S., and CHAN, T. Weighted essentially non-oscillatory schemes. Journal of Computational Physics, 115(1), 200-212(1994)
[2] JIANG, G. S. and SHU, C. W. Efficient implementation of weighted ENO schemes. Journal of Computational Physics, 126(1), 202-228(1996)
[3] TAYLOR, E. M., WU, M., and MARTÍN M. P. Optimization of nonlinear error for weighted essentially non-oscillatory methods in direct numerical simulations of compressible turbulence. Journal of Computational Physics, 223(1), 384-397(2007)
[4] JOHNSEN, E., LARSSON, J., BHAGATWALA, A. V., CABOT, W. H., MOIN, P., OLSON, B. J., RAWAT, P. S., SHANKAR, S. K., SJÖGREEN, B., YEE, H. C., ZHONG, X., and LELE, S. K. Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves. Journal of Computational Physics, 229(4), 1213-1237(2010)
[5] LELE, S. K. Compact finite difference schemes with spectral-like resolution. Journal of Computational Physics, 103(1), 16-42(1992)
[6] TAM, C. K. W. and WEBB, J. C. Dispersion-relation-preserving finite difference schemes for computational acoustics. Journal of Computational Physics, 107(2), 262-281(1993)
[7] LOCKARD, D. P., BRENTNER, K. S., and ATKINS, H. L. High-accuracy algorithms for computational aeroacoustics. AIAA Journal, 33(2), 246-251(1995)
[8] KIM, J. W. and LEE, D. J. Optimized compact finite difference schemes with maximum resolution. AIAA Journal, 34(5), 887-893(1996)
[9] WEIRS, V. and CANDLER, G. Optimization of weighted ENO schemes for DNS of compressible turbulence. 13th Computational Fluid Dynamics Conference, American Institute of Aeronautics and Astronautics, Reston (1997)
[10] MARTÍN, M. P., TAYLOR, E. M., WU, M., and WEIRS, V. G. A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence. Journal of Computational Physics, 220(1), 270-289(2006)
[11] ARSHED, G. M. and HOFFMANN, K. A. Minimizing errors from linear and nonlinear weights of WENO scheme for broadband applications with shock waves. Journal of Computational Physics, 246, 58-77(2013)
[12] WANG, Z. J. and CHEN, R. F. Optimized weighted essentially nonoscillatory schemes for linear waves with discontinuity. Journal of Computational Physics, 174(1), 381-404(2001)
[13] HILL, D. J. and PULLIN, D. I. Hybrid tuned center-difference-WENO method for large eddy simulations in the presence of strong shocks. Journal of Computational Physics, 194(2), 435-450(2004)
[14] HU, X. Y., TRITSCHLER, V. K., PIROZZOLI, S., and ADAMS, N. A. Dispersion-dissipation condition for finite difference schemes. arXiv, 1204.5088(2012) https://arxiv.org/abs/1204.5088
[15] FU, L., HU, X. Y., and ADAMS, N. A. A family of high-order targeted ENO schemes for compressible-fluid simulations. Journal of Computational Physics, 305, 333-359(2016)
[16] FU, L., HU, X. Y., and ADAMS, N. A. Targeted ENO schemes with tailored resolution property for hyperbolic conservation laws. Journal of Computational Physics, 349, 97-121(2017)
[17] DENG, X. and MAO, M. Weighted compact high-order nonlinear schemes for the Euler equations. 13th Computational Fluid Dynamics Conference, American Institute of Aeronautics and Astronautics, Reston (1997)
[18] DENG, X. and ZHANG, H. Developing high-order weighted compact nonlinear schemes. Journal of Computational Physics, 165(1), 22-44(2000)
[19] DENG, X., MAO, M., JIANG, Y., and LIU, H. New high-order hybrid cell-edge and cell-node weighted compact nonlinear schemes. 20th AIAA Computational Fluid Dynamics Conference, American Institute of Aeronautics and Astronautics, Reston (2011)
[20] DENG, X., JIANG, Y., MAO, M., LIU, H., and TU, G. Developing hybrid cell-edge and cell-node dissipative compact scheme for complex geometry flows. Science China Technological Sciences, 56(10), 2361-2369(2013)
[21] DENG, X., JIANG, Y., MAO, M., LIU, H., LI, S., and TU, G. A family of hybrid cell-edge and cell-node dissipative compact schemes satisfying geometric conservation law. Computers and Fluids, 116, 29-45(2015)
[22] NONOMURA, T. and FUJII, K. Robust explicit formulation of weighted compact nonlinear scheme. Computers and Fluids, 85, 8-18(2013)
[23] DENG, X., LIU, X., MAO, M., and ZHANG, H. Investigation on weighted compact fifth-order nonlinear scheme and applications to complex flow. 17th AIAA Computational Fluid Dynamics Conference, American Institute of Aeronautics and Astronautics, Reston (2005)
[24] NONOMURA, T. and FUJII, K. Effects of difference scheme type in high-order weighted compact nonlinear schemes. Journal of Computational Physics, 228(10), 3533-3539(2009)
[25] ZHANG, H., ZHANG, F., and XU, C. Towards optimal high-order compact schemes for simulating compressible flows. Applied Mathematics and Computation, 355, 221-237(2019)
[26] ZHANG, H., ZHANG, F., LIU, J., MCDONOUGH, J. M., and XU, C. A simple extended compact nonlinear scheme with adaptive dissipation control. Communications in Nonlinear Science and Numerical Simulation, 84, 105191(2020)
[27] YAMALEEV, N. K. and CARPENTER, M. H. Third-order energy stable WENO scheme. Journal of Computational Physics, 228(8), 3025-3047(2009)
[28] DON, W. S. and BORGES, R. Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes. Journal of Computational Physics, 250, 347-372(2013)
[29] JIA, F., GAO, Z., and DON, W. S. A spectral study on the dissipation and dispersion of the WENO schemes. Journal of Scientific Computing, 63(1), 49-77(2015)
[30] ZHENG, S., DENG, X., WANG, D., and XIE, C. A parameter-free ε-adaptive algorithm for improving weighted compact nonlinear schemes. International Journal for Numerical Methods in Fluids, 90(5), 247-266(2019)
[31] SHU, C. W. and OSHER, S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics, 77(2), 439-471(1988)
[32] ZHUANG, M. and CHEN, R. F. Optimized upwind dispersion-relation-preserving finite difference scheme for computational aeroacoustics. AIAA Journal, 36(11), 2146-2148(1998)
[33] LIN, S. Y. and HU, J. J. Parametric study of weighted essentially nonoscillatory schemes for computational aeroacoustics. AIAA Journal, 39(3), 371-379(2001)
[34] KIM, J. W. Optimised boundary compact finite difference schemes for computational aeroacoustics. Journal of Computational Physics, 225(1), 995-1019(2007)
[35] PIROZZOLI, S. On the spectral properties of shock-capturing schemes. Journal of Computational Physics, 219(2), 489-497(2006)
[36] ZHANG, S. and SHU, C. W. A new smoothness indicator for the WENO schemes and its effect on the convergence to steady state solutions. Journal of Scientific Computing, 31(1-2), 273-305(2007)
[37] ZHANG, S., JIANG, S., and SHU, C. W. Improvement of convergence to steady state solutions of Euler equations with the WENO schemes. Journal of Scientific Computing, 47(2), 216-238(2011)
[38] ZHANG, S., DENG, X., MAO, M., and SHU, C. W. Improvement of convergence to steady state solutions of Euler equations with weighted compact nonlinear schemes. Acta Mathematicae Applicatae Sinica, English Series, 29(3), 449-464(2013)
[39] WANG, D., DENG, X., WANG, G., and DONG, Y. Developing a hybrid flux function suitable for hypersonic flow simulation with high-order methods. International Journal for Numerical Methods in Fluids, 81(5), 309-327(2016)
[40] LAX, P. D. Weak solutions of nonlinear hyperbolic equations and their numerical computation. Communications on Pure and Applied Mathematics, 7(1), 159-193(1954)
[41] SOD, G. A. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. Journal of Computational Physics, 27(1), 1-31(1978)
[42] TORO, E. F. Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Springer-Verlag, Berlin (2009)
[43] SHU, C. W. and OSHER, S. Efficient implementation of essentially non-oscillatory shock-capturing schemes II. Journal of Computational Physics, 83(1), 32-78(1989)
[44] TITAREV, V. A. and TORO, E. F. Finite-volume WENO schemes for three-dimensional conservation laws. Journal of Computational Physics, 201(1), 238-260(2004)
[45] SHU, C. W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Springer, Berlin/Heidelberg, 325-432(1998)
[46] WOODWARD, P. and COLELLA, P. The numerical simulation of two-dimensional fluid flow with strong shocks. Journal of Computational Physics, 54(1), 115-173(1984)
[47] VAN REES, W. M., LEONARD, A., PULLIN, D. I., and KOUMOUTSAKOS, P. A comparison of vortex and pseudo-spectral methods for the simulation of periodic vortical flows at high Reynolds numbers. Journal of Computational Physics, 230(8), 2794-2805(2011)
[48] ZHANG, H., WANG, G., and ZHANG, F. A multi-resolution weighted compact nonlinear scheme for hyperbolic conservation laws. International Journal of Computational Fluid Dynamics, 34(3), 187-203(2020)