一种提高极值点处收敛精度的三阶WENO-Z格式
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摘要
为了提高三阶WENO-Z格式在极值点处的计算精度,通过理论推导给出三阶WENO格式满足收敛精度的充分条件。采用泰勒级数展开的方式,推导给出所构造格式非线性权重的计算公式,并综合权衡计算精度和计算稳定性确定所构造格式的参数。通过两个典型的精度测试,验证了改进格式在光滑流场极值点区域逼近三阶精度。进一步选用激波与熵波相互作用和Richtmyer-Meshkov不稳定性等经典算例,证实了本文提出的改进格式WENO-PZ3相较其他格式(WENO-JS3和WENO-Z3)不仅具有较高的精度,而且降低了格式的耗散,提高了对流场结构的分辨率。
In order to improve the convergence of the conventional third-order WENO-Z scheme at the critical points,the sufficient conditions for satisfying the convergence of the third-order WENO scheme are firstly derived.The expressions of the non-linear weights are derived from a Taylor series expansion.Then the parameters of the constructed scheme are finally determined considering the balance between the convergence precision and the computational stability.The accuracy tests prove that the proposed scheme almost converges to the third-order precision in a smooth flow field near the critical points.Shock-entropy wave test,Richtmyer-Meshkov instability and some other classic examples are computed to verify that the improved scheme WENO-PZ3 can give more accurate and higher-resolution results of the complex flow field structures compared with other WENO schemes like the WENO-JS3,and WENO-Z3.
In order to improve the convergence of the conventional third-order WENO-Z scheme at the critical points,the sufficient conditions for satisfying the convergence of the third-order WENO scheme are firstly derived.The expressions of the non-linear weights are derived from a Taylor series expansion.Then the parameters of the constructed scheme are finally determined considering the balance between the convergence precision and the computational stability.The accuracy tests prove that the proposed scheme almost converges to the third-order precision in a smooth flow field near the critical points.Shock-entropy wave test,Richtmyer-Meshkov instability and some other classic examples are computed to verify that the improved scheme WENO-PZ3 can give more accurate and higher-resolution results of the complex flow field structures compared with other WENO schemes like the WENO-JS3,and WENO-Z3.
引文
[1] Liu X D,Osher S,Chan T.Weighted essentially non-oscillatory schemes [J].Journal of Computational Physics,1994,115(1):200-212.
[2] Harten A,Engquist B,Osher S,et al.Uniformly High Order Accurate Essentially Non-oscillatory Schemes,III[M].Springer Berlin Heidelberg,1987.
[3] Jiang G S,Shu C W.Efficient implementation of weighted ENO schemes [J].Journal of Computational Physics,1995,126(1):202-228.
[4] Hsieh T J,Wang C H,Yang J Y.Numerical experiments with several variant WENO schemes for the Euler equations [J].International Journal for Numerical Methods in Fluids,2008,58(9):1017-1039.
[5] Zhao S,Lardjane N,Fedioun I.Comparison of improved finite -difference WENO schemes for the implicit large eddy simulation of turbulent non-reacting and reacting high-speed shear flows [J].Computers & Fluids,2014,95:74-87.
[6] Wang C,Ding J X,Shu C W,et al.Three -dimensional ghost-fluid large -scale numerical investigation on air explosion [J].Computers & Fluids,2016,137:70-79.
[7] Zaghi S,Mascio A D,Favini B.Application of WENO -positivity -preserving schemes to highly under-expanded jets [J].Journal of Scientific Computing,2016,69(3):1033-1057.
[8] Henrick A K,Aslam T D,Powers J M.Mapped weighted essentially non-oscillatory schemes:Achieving optimal order near critical points [J].Journal of Computational Physics,2005,207(2):542-567.
[9] Borges R,Carmona M,Costa B,et al.An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws [J].Journal of Computational Physics,2008,227(6):3191-3211.
[10] Yamaleev N K,Carpenter M H.A systematic metho -dology for constructing high-order energy stable WENO schemes [J].Journal of Computational Physics,2009,228(11):4248-4272.
[11] Fan P.High order weighted essentially nonoscillatory WENO -η schemes for hyperbolic conservation laws [J].Journal of Computational Physics,2014,269:355-385.
[12] Fan P,Shen Y Q,Tian B L,et al.A new smoothness indicator for improving the weighted essentially non-oscillatory scheme [J].Journal of Computational Physics,2014,269:329-354
[13] Feng H,Huang C,Wang R.An improved mapped weighted essentially non-oscillatory scheme[J].Applied Mathematics and Computation,2014,232:453-468.
[14] Shen Y Q,Zha G C.Improvement of weighted essentially non-oscillatory schemes near discontinuities [J].Computers & Fluids,2014,96:1-9.
[15] Kim C H,Ha Y,Yoon J.Modified non-linear weights for fifth-order weighted essentially non-oscillatory schemes [J].Journal of Scientific Computing,2016,67(1):299-323.
[16] Ma Y G,Yan Z G,Zhu H J.Improvement of multistep WENO scheme and its extension to higher orders of accuracy [J].International Journal for Numerical Methods,2016,82(12):818-838.
[17] Wang R,Feng H,Huang C.A new mapped weighted essentially non-oscillatory method using rational mapping function [J].Journal of Scientific Computing,2016,67(2):540-580.
[18] Yamaleev N K,Carpenter M H.Third-order energy stable WENO scheme [J].Journal of Computational Physics,2013,228(8):3025-3047.
[19] Wu X S,Zhao Y X.A high-resolution hybrid scheme for hyperbolic conservation laws [J].International Journal for Numerical Methods in Fluids,2015,78(3):162-187.
[20] Don W S,Borges R.Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes [J].Journal of Computational Physics,2013,250:347-372.
[21] Hu X Y,Wang Q,Adams N A.An adaptive central-upwind weighted essentially non-oscillatory scheme [J].Journal of Computational Physics,2010,229:8952-8965.
[22] Wu X S,Liang J H,Zhao Y X.A new smoothness indicator for third-order WENO scheme [J].International Journal for Numerical Methods in Fluids,2016,81(7):451-459.
[23] Gande N R,Rathod Y,Rathan S.Third-order WENO scheme with a new smoothness indicator [J].International Journal for Numerical Methods in Fluids,2017,85(2):171-185.
[24] 徐维铮,孔祥韶,郑成,等.三阶WENO格式精度分析与改进 [J].计算力学学报,2018,35(1):51-55.(XUWei-zheng,KONG Xiang-shao,ZHENG Cheng,et al.Precision analysis and improvement of third-order WENO scheme[J].Chinese Journal of Computational Mechanics,2018,35(1):51-55.(in Chinese))
[25] Shu C W,Osher S.Efficient implementation of essentially non-oscillatory shock-capturing schemes,II [J].Journal of Computational Physics,1988,77(2):439-471.
[26] Sod G A.A survey of several finite difference me -thods for systems of nonlinear hyperbolic conservation laws [J].Journal of Computational Physics,1978,27(1):1-31.
[27] Titarev V A,Toro E F.Finite -volume WENO schemes for three-dimensional conservation laws [J].Journal of Computational Physics,2004,201(1):238-260.
[28] Zhang P G,Wang J P.A newly improved WENO scheme and its application to the simulation of richtmyer-meshkov instability [J].Procedia Engineering,2013,61:325-332.
[29] Shi J,Zhang Y T,Shu C W.Resolution of high order WENO schemes for complicated flow structures [J].Journal of Computational Physics,2003,186(2):690-696.
[30] Acker F,Borges R,Costa B.An improved WENO -Z scheme [J].Journal of Computational Physics,2016,313:726-753.
[2] Harten A,Engquist B,Osher S,et al.Uniformly High Order Accurate Essentially Non-oscillatory Schemes,III[M].Springer Berlin Heidelberg,1987.
[3] Jiang G S,Shu C W.Efficient implementation of weighted ENO schemes [J].Journal of Computational Physics,1995,126(1):202-228.
[4] Hsieh T J,Wang C H,Yang J Y.Numerical experiments with several variant WENO schemes for the Euler equations [J].International Journal for Numerical Methods in Fluids,2008,58(9):1017-1039.
[5] Zhao S,Lardjane N,Fedioun I.Comparison of improved finite -difference WENO schemes for the implicit large eddy simulation of turbulent non-reacting and reacting high-speed shear flows [J].Computers & Fluids,2014,95:74-87.
[6] Wang C,Ding J X,Shu C W,et al.Three -dimensional ghost-fluid large -scale numerical investigation on air explosion [J].Computers & Fluids,2016,137:70-79.
[7] Zaghi S,Mascio A D,Favini B.Application of WENO -positivity -preserving schemes to highly under-expanded jets [J].Journal of Scientific Computing,2016,69(3):1033-1057.
[8] Henrick A K,Aslam T D,Powers J M.Mapped weighted essentially non-oscillatory schemes:Achieving optimal order near critical points [J].Journal of Computational Physics,2005,207(2):542-567.
[9] Borges R,Carmona M,Costa B,et al.An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws [J].Journal of Computational Physics,2008,227(6):3191-3211.
[10] Yamaleev N K,Carpenter M H.A systematic metho -dology for constructing high-order energy stable WENO schemes [J].Journal of Computational Physics,2009,228(11):4248-4272.
[11] Fan P.High order weighted essentially nonoscillatory WENO -η schemes for hyperbolic conservation laws [J].Journal of Computational Physics,2014,269:355-385.
[12] Fan P,Shen Y Q,Tian B L,et al.A new smoothness indicator for improving the weighted essentially non-oscillatory scheme [J].Journal of Computational Physics,2014,269:329-354
[13] Feng H,Huang C,Wang R.An improved mapped weighted essentially non-oscillatory scheme[J].Applied Mathematics and Computation,2014,232:453-468.
[14] Shen Y Q,Zha G C.Improvement of weighted essentially non-oscillatory schemes near discontinuities [J].Computers & Fluids,2014,96:1-9.
[15] Kim C H,Ha Y,Yoon J.Modified non-linear weights for fifth-order weighted essentially non-oscillatory schemes [J].Journal of Scientific Computing,2016,67(1):299-323.
[16] Ma Y G,Yan Z G,Zhu H J.Improvement of multistep WENO scheme and its extension to higher orders of accuracy [J].International Journal for Numerical Methods,2016,82(12):818-838.
[17] Wang R,Feng H,Huang C.A new mapped weighted essentially non-oscillatory method using rational mapping function [J].Journal of Scientific Computing,2016,67(2):540-580.
[18] Yamaleev N K,Carpenter M H.Third-order energy stable WENO scheme [J].Journal of Computational Physics,2013,228(8):3025-3047.
[19] Wu X S,Zhao Y X.A high-resolution hybrid scheme for hyperbolic conservation laws [J].International Journal for Numerical Methods in Fluids,2015,78(3):162-187.
[20] Don W S,Borges R.Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes [J].Journal of Computational Physics,2013,250:347-372.
[21] Hu X Y,Wang Q,Adams N A.An adaptive central-upwind weighted essentially non-oscillatory scheme [J].Journal of Computational Physics,2010,229:8952-8965.
[22] Wu X S,Liang J H,Zhao Y X.A new smoothness indicator for third-order WENO scheme [J].International Journal for Numerical Methods in Fluids,2016,81(7):451-459.
[23] Gande N R,Rathod Y,Rathan S.Third-order WENO scheme with a new smoothness indicator [J].International Journal for Numerical Methods in Fluids,2017,85(2):171-185.
[24] 徐维铮,孔祥韶,郑成,等.三阶WENO格式精度分析与改进 [J].计算力学学报,2018,35(1):51-55.(XUWei-zheng,KONG Xiang-shao,ZHENG Cheng,et al.Precision analysis and improvement of third-order WENO scheme[J].Chinese Journal of Computational Mechanics,2018,35(1):51-55.(in Chinese))
[25] Shu C W,Osher S.Efficient implementation of essentially non-oscillatory shock-capturing schemes,II [J].Journal of Computational Physics,1988,77(2):439-471.
[26] Sod G A.A survey of several finite difference me -thods for systems of nonlinear hyperbolic conservation laws [J].Journal of Computational Physics,1978,27(1):1-31.
[27] Titarev V A,Toro E F.Finite -volume WENO schemes for three-dimensional conservation laws [J].Journal of Computational Physics,2004,201(1):238-260.
[28] Zhang P G,Wang J P.A newly improved WENO scheme and its application to the simulation of richtmyer-meshkov instability [J].Procedia Engineering,2013,61:325-332.
[29] Shi J,Zhang Y T,Shu C W.Resolution of high order WENO schemes for complicated flow structures [J].Journal of Computational Physics,2003,186(2):690-696.
[30] Acker F,Borges R,Costa B.An improved WENO -Z scheme [J].Journal of Computational Physics,2016,313:726-753.