计算动力学中的伪弧长方法研究
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摘要
动力学问题通常采用微分方程来描绘,但由于工程实际问题的复杂性,微分方程模型常伴随着解的不连续性、刚性或激波间断奇异性特点,传统方法很难求解,奇异性问题是计算动力学难点,同时也是国内外学者研究的热点.伪弧长数值算法是针对计算动力学中的奇异性问题所提出的,其基本思想为通过在解曲线上引入伪弧长参数,并增加一个约束方程,在伪弧长参数作用下,使得原始离散单元发生扭曲形变,从而达到消除或减弱奇异性的目的.本文首先介绍伪弧长方法求解定常对流-扩散方程的奇异性问题,并提出针对双曲守恒定律的局部伪弧长算法,其思想在于首先通过间断解的梯度变换来确定强间断所处位置,进而通过局部网格点重构以及数值修正来达到强间断处奇异性消除与降低的目的.针对高维问题,提出全局伪弧长方法,通过对整个计算区域内的网格点进行重构,使得所有网格点向奇异间断点处移动,从而降低间断点的影响域,达到降低奇异性的目的.重点讨论了三维全局伪弧长算法问题的计算难点,即三维空间网格扭曲大变形导致的数值算法不收敛,并提出在算法设计过程中采用分块重构与整体计算相结合的策略,实现了三维空间中的伪弧长数值算法,最后通过数值实验来验证伪弧长算法对于奇异性问题的有效性.
Differential equations are often used to describe the dynamic problems. Classical approaches are always hard to solve it in engineering practice due to its characteristics of strong discontinuity, rigidity and shock singularity,among which singularity problem is one of the most difficult and hot issues among scholars. Pseudo arc-length numerical algorithm is proposed for singularity problems in computational dynamics, whose basic idea is to introduce a pseudo arc-length parameter in the original governing equations so that a constraint equation is added. Under the action of a pseudo arc-length parameter, the original discrete elements are distorted to achieve the goal of eliminating or weakening singularity. Firstly, this paper gave an introduction about the pseudo arc-length method for solving the singularity problem in steady diffusion-convection equations. Then the local pseudo arc-length algorithm is proposed for hyperbolic conservation laws. This algorithm has two steps. The first step is to determine the location of the strong discontinuity and the second one aims to eliminate or reduce the singularity by reconstructing the local mesh. Secondly, a global pseudoarc-length algorithm is put forward for high dimensional problems, which can reconstruct the mesh in whole area. Since the reconstructed mesh can adaptively catch the singularity points, the singularity is greatly reduced. Thirdly, the threedimensional global pseudo arc-length algorithm and its computational difficulties in numerical algorithm convergence caused by large grid distortion in three-dimensional space are presented. Then the combination of block reconstruction and integral calculation strategy is adopted in the algorithm design process to achieve the pseudo arc-length numerical algorithm in three-dimensional space. Finally, numerical examples were employed to verify the validity of the pseudo arc-length algorithm.
Differential equations are often used to describe the dynamic problems. Classical approaches are always hard to solve it in engineering practice due to its characteristics of strong discontinuity, rigidity and shock singularity,among which singularity problem is one of the most difficult and hot issues among scholars. Pseudo arc-length numerical algorithm is proposed for singularity problems in computational dynamics, whose basic idea is to introduce a pseudo arc-length parameter in the original governing equations so that a constraint equation is added. Under the action of a pseudo arc-length parameter, the original discrete elements are distorted to achieve the goal of eliminating or weakening singularity. Firstly, this paper gave an introduction about the pseudo arc-length method for solving the singularity problem in steady diffusion-convection equations. Then the local pseudo arc-length algorithm is proposed for hyperbolic conservation laws. This algorithm has two steps. The first step is to determine the location of the strong discontinuity and the second one aims to eliminate or reduce the singularity by reconstructing the local mesh. Secondly, a global pseudoarc-length algorithm is put forward for high dimensional problems, which can reconstruct the mesh in whole area. Since the reconstructed mesh can adaptively catch the singularity points, the singularity is greatly reduced. Thirdly, the threedimensional global pseudo arc-length algorithm and its computational difficulties in numerical algorithm convergence caused by large grid distortion in three-dimensional space are presented. Then the combination of block reconstruction and integral calculation strategy is adopted in the algorithm design process to achieve the pseudo arc-length numerical algorithm in three-dimensional space. Finally, numerical examples were employed to verify the validity of the pseudo arc-length algorithm.
引文
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2杨露斯,黎炼.论计算机发展史及展望.信息与电脑,2010,6(1):188-188(Yang Lusi,Li Lian.Theory of computer development and prospects.China Computer&Communication,2010,6(1):188-188(in Chinese))
3 Luo ST,Tran HV,Yu Y.Some inverse problems in periodic homogenization of hamilton-jacobi equations.Archive for Rational Mechanics and Analysis,2016,221(3):1585-1617
4 Deleuze Y,Chiang CY,Thiriet M,et al.Numerical study of plume patterns in a chemotaxis-diffusion-convection coupling system.Computers&Fluids,2016,126(1):58-70
5 Tian DS.Multiple positive periodic solutions for second-order differential equations with a singularity.Acta Applicandae Mathematicae,2016,144(1):1-10
6 Riks E.An incremental approach to the solution of snapping and buckling problems.International Journal of Solids&Structures,1979,15(7):529-551
7 Boor CD.On local spline approximation by moments.Journal of Mathematics and Mechanics,1968,17(1):729-735
8 Crisfield MA.An arc-length method including line searches and accelerations.International Journal for Numerical Methods in Engineering,1983,19(1):1268-1289
9 Chan TF.Newton-like pseudo-arclength methods for computing simple turning points.Siam Journal on Scientific&Statistical Computing,1984,5(1):135-148
10忻孝康,唐登海.一维定常对流-扩散方程的伪弧长参数法.应用力学学报,1988,5(1):45-54(Xin Xiaokang,Tang Denghai.An arc-length parameter technique for steady diffusion-convection equation.Chinese Journal of Applied Mechanics,1988,5(1):45-54(in Chinese))
11陈国谦,高智.对流扩散方程的迎风变换及相应有限差分方法.力学学报,1991,23(4):418-425(Chen Guoqian,Gao Zhi.Transformation of the convective diffusion equation with corresponding finite difference method.Chinese Journal of Theoretical and Applied Mechanics,1991,23(4):418-425(in Chinese))
12武际可,许为厚,丁红丽.求解微分方程初值问题的一种弧长法.应用数学和力学,1999,20(8):875-880(Wu Jike,Xu Weihou,Ding Hongli.Arc-length method for differential equations.Applied Mathematics and Mechanics,1999,20(8):875-880(in Chinese))
13 Harten A,Hyman JM.Self adjusting grid methods for onedimensional hyperbolic conservation laws.Journal of Computational Physics,1983,50(2):235-269
14 Ren YH,Russell RD.Moving mesh techniques based upon equidistribution,and their stability.Siam Journal on Scientific&Statistical Computing,1992,13(6):1265-1286
15 Huang WZ,Ren YH,Russell RD.Moving mesh partial differential equations(MMPDEs)based upon the equidistribution principle.Siam Journal on Numerical Analysis,1994,31(3):709-730
16 White AB.On selection of equidistributing meshes for two-point boundary-value problems.Siam Journal on Numerical Analysis,1979,16(3):472-502
17 Stockie JM,Mackenzie JA,Russell RD.A moving mesh method for one-dimensional hyperbolic conservation laws.Siam Journal on Scientific Computing,2001,22(5):1791-1813
18 Tang HZ,Tang T.Adaptive mesh methods for one-and twodimensional hyperbolic conservation laws.Siam Journal on Numerical Analysis,2003,41(2):487-515
19 Ning JG,Wang X,Ma TB,et al.Numerical simulation of shock wave interaction with a deformable particle based on the pseudo arclength method.Science China Technological Sciences,2015,58(5):848-857
20王星,马天宝,宁建国.双曲偏微分方程的局部伪弧长方法研究.计算力学学报,2014,31(3):384-389(Wang Xing,Ma Tianbao,Ning Jianguo.Local pseudo arc-length method for hyperbolic partial differential equation.Chinese Journal of Computational Mechanics,2014,31(3):384-389(in Chinese))
21 Wang X,Ma TB,Ning JG.A pseudo arc-length method for numerical simulation of shock waves.Chinese Physics Letter,2014,31(3):030201-030201
22 Wang X,Ma TB,Ren HL,et al.A local pseudo arc-length method for hyperbolic conservation laws.Acta Mechanica Sinica,2014,30(6):956-965
23 Ning JG,Yuan XP,Ma TB,et al.Positivity-preserving moving mesh scheme for two-step reaction model in two dimensions.Computers&Fluids,2015,123(1):72-86
24 Yuan XP,Ning JG,Ma TB,et al.Stability of newton tvd runge-kutta scheme for one-dimensional Euler equations with adaptive mesh.Applied Mathematics&Computation,2016,282(1):1-16
25 Sun LP,Cong YH,Kuang JX.Asymptotic behavior of nonlinear delay differential-algebraic equations and implicit Euler methods.Applied Mathematics&Computation,2014,228(1):395-403
26 Harwood SM,Kai H,Barton PI.Efficient solution of ordinary differential equations with a parametric lexicographic linear program embedded.Numerische Mathematik,2016,133(4):623-653
27 Perthame B,Shu CW.On positivity preserving finite volume schemes for Euler equations.Numerische Mathematik,1996,73(1):119-130
28 Wang C,Zhang X,Shu CW,et al.Robust high order discontinuous galerkin schemes for two-dimensional gaseous detonations.Journal of Computational Physics,2012,231(2):653-665
29魏涛,许明田,汪引.求解对流扩散问题的积分方程法.化工学报,2015,66(10):3888-3894(Wei Tao,Xu Mingtian,Wang Yin.Integral equation approach to convection-diffusion problems.Ciesc Journal,2015,66(10):3888-3894(in Chinese))
30 Zhou XF,Guo J,Li F.Stability,accuracy and numerical diffusion analysis of nodal expansion method for steady convection diffusion equation.Nuclear Engineering&Design,2015,295(1):567-575
31 Sandu A.Rosenbrock methods with an explicit first stage.International Journal of Computer Mathematics,2015,93(6):1-18
32冯帆,王自发,唐晓.一个基于打靶法的大气污染源反演自适应算法.大气科学,2016,40(4):719-729(Feng Fan,Wang Zifa,Tang Xiao.Development of an adaptive algorithm based on the shooting method and its application in the problem of estimating air pollutant emissions.Chinese Journal of Atmospheric Sciences,2016,40(4):719-729(in Chinese))
33 Chen Z,Huang H,Yan J.Third order maximum-principle-satisfying direct discontinuous galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes.Journal of Computational Physics,2015,308(1):198-217
34刘朝阳,王振国,孙明波,等.一种求解激波问题的中心差分-WENO混合方法研究.推进技术,2016,37(8):1522-1528(Liu Chaoyang,Wang Zhenguo,Sun Mingbo,et al.Investigation of a hybrid central difference-WENO method for shock wave problems.Journal of Propulsion Technology,2016,37(8):1522-1528(in Chinese))
35 Shu CW,Osher S.Efficient implementation of essentially nonoscillatory shock-capturing schemes.Journal of Computational Physics,1989,83(1):32-78
36 Gotlieb S,Shu CW.Total variation diminishing Runge-Kutta schemes.Mathematics of Computation,1996,67(221):73-85
37 Doring W.On detonation processes in gases.Annals of Physics,1943,43(9):421-436
38 Fickett W,Wood WW.Flow calculations for pulsating onedimensional detonations.The Physics of Fluids,1966,9(5):903-916
2杨露斯,黎炼.论计算机发展史及展望.信息与电脑,2010,6(1):188-188(Yang Lusi,Li Lian.Theory of computer development and prospects.China Computer&Communication,2010,6(1):188-188(in Chinese))
3 Luo ST,Tran HV,Yu Y.Some inverse problems in periodic homogenization of hamilton-jacobi equations.Archive for Rational Mechanics and Analysis,2016,221(3):1585-1617
4 Deleuze Y,Chiang CY,Thiriet M,et al.Numerical study of plume patterns in a chemotaxis-diffusion-convection coupling system.Computers&Fluids,2016,126(1):58-70
5 Tian DS.Multiple positive periodic solutions for second-order differential equations with a singularity.Acta Applicandae Mathematicae,2016,144(1):1-10
6 Riks E.An incremental approach to the solution of snapping and buckling problems.International Journal of Solids&Structures,1979,15(7):529-551
7 Boor CD.On local spline approximation by moments.Journal of Mathematics and Mechanics,1968,17(1):729-735
8 Crisfield MA.An arc-length method including line searches and accelerations.International Journal for Numerical Methods in Engineering,1983,19(1):1268-1289
9 Chan TF.Newton-like pseudo-arclength methods for computing simple turning points.Siam Journal on Scientific&Statistical Computing,1984,5(1):135-148
10忻孝康,唐登海.一维定常对流-扩散方程的伪弧长参数法.应用力学学报,1988,5(1):45-54(Xin Xiaokang,Tang Denghai.An arc-length parameter technique for steady diffusion-convection equation.Chinese Journal of Applied Mechanics,1988,5(1):45-54(in Chinese))
11陈国谦,高智.对流扩散方程的迎风变换及相应有限差分方法.力学学报,1991,23(4):418-425(Chen Guoqian,Gao Zhi.Transformation of the convective diffusion equation with corresponding finite difference method.Chinese Journal of Theoretical and Applied Mechanics,1991,23(4):418-425(in Chinese))
12武际可,许为厚,丁红丽.求解微分方程初值问题的一种弧长法.应用数学和力学,1999,20(8):875-880(Wu Jike,Xu Weihou,Ding Hongli.Arc-length method for differential equations.Applied Mathematics and Mechanics,1999,20(8):875-880(in Chinese))
13 Harten A,Hyman JM.Self adjusting grid methods for onedimensional hyperbolic conservation laws.Journal of Computational Physics,1983,50(2):235-269
14 Ren YH,Russell RD.Moving mesh techniques based upon equidistribution,and their stability.Siam Journal on Scientific&Statistical Computing,1992,13(6):1265-1286
15 Huang WZ,Ren YH,Russell RD.Moving mesh partial differential equations(MMPDEs)based upon the equidistribution principle.Siam Journal on Numerical Analysis,1994,31(3):709-730
16 White AB.On selection of equidistributing meshes for two-point boundary-value problems.Siam Journal on Numerical Analysis,1979,16(3):472-502
17 Stockie JM,Mackenzie JA,Russell RD.A moving mesh method for one-dimensional hyperbolic conservation laws.Siam Journal on Scientific Computing,2001,22(5):1791-1813
18 Tang HZ,Tang T.Adaptive mesh methods for one-and twodimensional hyperbolic conservation laws.Siam Journal on Numerical Analysis,2003,41(2):487-515
19 Ning JG,Wang X,Ma TB,et al.Numerical simulation of shock wave interaction with a deformable particle based on the pseudo arclength method.Science China Technological Sciences,2015,58(5):848-857
20王星,马天宝,宁建国.双曲偏微分方程的局部伪弧长方法研究.计算力学学报,2014,31(3):384-389(Wang Xing,Ma Tianbao,Ning Jianguo.Local pseudo arc-length method for hyperbolic partial differential equation.Chinese Journal of Computational Mechanics,2014,31(3):384-389(in Chinese))
21 Wang X,Ma TB,Ning JG.A pseudo arc-length method for numerical simulation of shock waves.Chinese Physics Letter,2014,31(3):030201-030201
22 Wang X,Ma TB,Ren HL,et al.A local pseudo arc-length method for hyperbolic conservation laws.Acta Mechanica Sinica,2014,30(6):956-965
23 Ning JG,Yuan XP,Ma TB,et al.Positivity-preserving moving mesh scheme for two-step reaction model in two dimensions.Computers&Fluids,2015,123(1):72-86
24 Yuan XP,Ning JG,Ma TB,et al.Stability of newton tvd runge-kutta scheme for one-dimensional Euler equations with adaptive mesh.Applied Mathematics&Computation,2016,282(1):1-16
25 Sun LP,Cong YH,Kuang JX.Asymptotic behavior of nonlinear delay differential-algebraic equations and implicit Euler methods.Applied Mathematics&Computation,2014,228(1):395-403
26 Harwood SM,Kai H,Barton PI.Efficient solution of ordinary differential equations with a parametric lexicographic linear program embedded.Numerische Mathematik,2016,133(4):623-653
27 Perthame B,Shu CW.On positivity preserving finite volume schemes for Euler equations.Numerische Mathematik,1996,73(1):119-130
28 Wang C,Zhang X,Shu CW,et al.Robust high order discontinuous galerkin schemes for two-dimensional gaseous detonations.Journal of Computational Physics,2012,231(2):653-665
29魏涛,许明田,汪引.求解对流扩散问题的积分方程法.化工学报,2015,66(10):3888-3894(Wei Tao,Xu Mingtian,Wang Yin.Integral equation approach to convection-diffusion problems.Ciesc Journal,2015,66(10):3888-3894(in Chinese))
30 Zhou XF,Guo J,Li F.Stability,accuracy and numerical diffusion analysis of nodal expansion method for steady convection diffusion equation.Nuclear Engineering&Design,2015,295(1):567-575
31 Sandu A.Rosenbrock methods with an explicit first stage.International Journal of Computer Mathematics,2015,93(6):1-18
32冯帆,王自发,唐晓.一个基于打靶法的大气污染源反演自适应算法.大气科学,2016,40(4):719-729(Feng Fan,Wang Zifa,Tang Xiao.Development of an adaptive algorithm based on the shooting method and its application in the problem of estimating air pollutant emissions.Chinese Journal of Atmospheric Sciences,2016,40(4):719-729(in Chinese))
33 Chen Z,Huang H,Yan J.Third order maximum-principle-satisfying direct discontinuous galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes.Journal of Computational Physics,2015,308(1):198-217
34刘朝阳,王振国,孙明波,等.一种求解激波问题的中心差分-WENO混合方法研究.推进技术,2016,37(8):1522-1528(Liu Chaoyang,Wang Zhenguo,Sun Mingbo,et al.Investigation of a hybrid central difference-WENO method for shock wave problems.Journal of Propulsion Technology,2016,37(8):1522-1528(in Chinese))
35 Shu CW,Osher S.Efficient implementation of essentially nonoscillatory shock-capturing schemes.Journal of Computational Physics,1989,83(1):32-78
36 Gotlieb S,Shu CW.Total variation diminishing Runge-Kutta schemes.Mathematics of Computation,1996,67(221):73-85
37 Doring W.On detonation processes in gases.Annals of Physics,1943,43(9):421-436
38 Fickett W,Wood WW.Flow calculations for pulsating onedimensional detonations.The Physics of Fluids,1966,9(5):903-916