三阶WENO-Z格式精度分析及其改进格式
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摘要
首先通过理论推导给出了三阶WENO格式(WENO-JS3格式)满足收敛精度的充分条件.采用Taylor(泰勒)级数展开的方法,分析发现传统的三阶WENO-Z格式(WENO-Z3格式)在光滑流场极值点处精度降低.为了提高WENO-Z3格式在极值点处的计算精度,根据收敛精度的充分条件构造一种改进的三阶WENO-Z格式(WENO-NZ3格式),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WENO-NZ3格式在光滑流场极值点区域逼近三阶精度.选用Sod激波管、激波与熵波相互作用、Rayleigh-Taylor不稳定性、二维Riemann(黎曼)问题经典算例,进一步证实了本文提出的WENO-NZ3格式相较其他格式(WENO-JS3、WENOZ3、WENO-N3),不仅提高了计算精度,而且提高了对复杂流场结构的分辨率.
Firstly,the sufficient conditions for the 3 rd-order WENO scheme satisfying the convergence precision were deduced. Based on the Taylor series method,the precision of the conventional 3 rd-order WENO-Z scheme in the smooth flow field was analyzed. It was found that at the critical points,the 3 rd-order WENO-Z scheme fails to achieve the convergence precision. In order to improve the precision near the critical points for the 3 rd-order WENO-Z scheme,an improved 3 rd-order WENO-Z scheme( WENONZ3) was constructed in view of the balance between precision and stability to finally determine the parameters. The improvement of the precision was verified through 2 typical numerical tests. What is more,the Sod shock wave tube,the shock-entropy wave interaction,the Rayleigh-Taylor instability and the 2 D Riemann problem were calculated to confirm that the WENO-NZ3 scheme performs better than the conventional WENO schemes like WENO-JS3,WENO-Z3 and WENO-N3.
Firstly,the sufficient conditions for the 3 rd-order WENO scheme satisfying the convergence precision were deduced. Based on the Taylor series method,the precision of the conventional 3 rd-order WENO-Z scheme in the smooth flow field was analyzed. It was found that at the critical points,the 3 rd-order WENO-Z scheme fails to achieve the convergence precision. In order to improve the precision near the critical points for the 3 rd-order WENO-Z scheme,an improved 3 rd-order WENO-Z scheme( WENONZ3) was constructed in view of the balance between precision and stability to finally determine the parameters. The improvement of the precision was verified through 2 typical numerical tests. What is more,the Sod shock wave tube,the shock-entropy wave interaction,the Rayleigh-Taylor instability and the 2 D Riemann problem were calculated to confirm that the WENO-NZ3 scheme performs better than the conventional WENO schemes like WENO-JS3,WENO-Z3 and WENO-N3.
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]//HUSSAINI M Y,VAN LEER B,VAN ROSENDALE J,ed.Upwind and High-Resolution Schemes.Berlin,Heidelberg:Springer,1987:218-290.
[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].Computes&Fluids,2014,95(3):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].Computes&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):1-25.
[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 methodology 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-ηschemes for hyperbolic conservation laws[J].Journal of Computational Physics,2014,269(1):355-385.
[12]FAN P,SHEN Y,TIAN B,et al.A new smoothness indicator for improving the weighted essentially non-oscillatory scheme[J].Journal of Computational Physics,2014,269(10):329-354.
[13]FENG H,HUANG C,WANG R.An improved mapped weighted essentially non-oscillatory scheme[J].Applied Mathematics&Computation,2014,232(6):453-468.
[14]SHEN Y,ZHA G.Improvement of weighted essentially non-oscillatory schemes near discontinuities[J].Computes&Fluids,2014,96(12):1-9.
[15]CHANG H K,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,YAN Z,ZHU H.Improvement of multistep WENO scheme and its extension to higher orders of accuracy[J].International Journal for Numerical Methods in Fluids,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 Xiaoshuai,ZHAO Yuxin.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(23):8952-8965.
[22]WU X,LIANG J,ZHAO Y.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]SHU C W,OSHER S.Efficient implementation of essentially non-oscillatory shock-capturing schemes,II[J].Journal of Computational Physics,1989,83(1):32-78.
[25]SOD G A.A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws[J].Journal of Computational Physics,1978,27(1):1-31.
[26]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.
[27]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.
[28]ACKER F,BORGES R,COSTA B.An improved WENO-Z scheme[J].Journal of Computational Physics,2016,313:726-753.
[29]LAX P D,LIU X D.Solution of two-dimensional Riemann problems of gas dynamics by positive schemes[J].SIAM Journal on Scientific Computing,1998,19(2):319-340.
[2]HARTEN A,ENGQUIST B,OSHER S,et al.Uniformly High Order Accurate Essentially Non-Oscillatory Schemes,III[M]//HUSSAINI M Y,VAN LEER B,VAN ROSENDALE J,ed.Upwind and High-Resolution Schemes.Berlin,Heidelberg:Springer,1987:218-290.
[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].Computes&Fluids,2014,95(3):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].Computes&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):1-25.
[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 methodology 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-ηschemes for hyperbolic conservation laws[J].Journal of Computational Physics,2014,269(1):355-385.
[12]FAN P,SHEN Y,TIAN B,et al.A new smoothness indicator for improving the weighted essentially non-oscillatory scheme[J].Journal of Computational Physics,2014,269(10):329-354.
[13]FENG H,HUANG C,WANG R.An improved mapped weighted essentially non-oscillatory scheme[J].Applied Mathematics&Computation,2014,232(6):453-468.
[14]SHEN Y,ZHA G.Improvement of weighted essentially non-oscillatory schemes near discontinuities[J].Computes&Fluids,2014,96(12):1-9.
[15]CHANG H K,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,YAN Z,ZHU H.Improvement of multistep WENO scheme and its extension to higher orders of accuracy[J].International Journal for Numerical Methods in Fluids,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 Xiaoshuai,ZHAO Yuxin.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(23):8952-8965.
[22]WU X,LIANG J,ZHAO Y.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]SHU C W,OSHER S.Efficient implementation of essentially non-oscillatory shock-capturing schemes,II[J].Journal of Computational Physics,1989,83(1):32-78.
[25]SOD G A.A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws[J].Journal of Computational Physics,1978,27(1):1-31.
[26]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.
[27]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.
[28]ACKER F,BORGES R,COSTA B.An improved WENO-Z scheme[J].Journal of Computational Physics,2016,313:726-753.
[29]LAX P D,LIU X D.Solution of two-dimensional Riemann problems of gas dynamics by positive schemes[J].SIAM Journal on Scientific Computing,1998,19(2):319-340.
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