基于GDQM的一维双曲守恒律数值方法研究
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摘要
双曲型守恒律方程数值计算方法的研究,从二十世纪三十年代以来,一直都是计算数学中的一个重要研究领域,并且出现了许多有效的数值计算方法。其中GDQ方法以其形式简单,原理浅显易懂,对网格的形状和数目都不作过多限制等诸多优点而被研究和推广。
本文所做的工作如下:
介绍了双曲型守恒律方程(组)及其解析解,阐述了离散GDQ方法产生的背景,针对一维双曲型守恒律方程(组)具有间断解(激波)的可压缩流问题,给出高精度、高分辨率的自适应离散GDQM和形式一致的离散GDQM,并详细分析了这两个格式的计算原理,给出了构造这两个格式的基本步骤,即网格剖分、空间离散、时间离散。其中空间离散的重构步是最关键的,它是在整个区间上离散并重构函数,这样处理使得计算效率有明显提高;时间离散采用经典方法——TVD型Runge-Kutta时间离散法。
这两种离散GDQ格式都是TVD的。
通过典型算例表明这两个数值方法都能很好地模拟间断,清晰地捕捉到激波、稀疏波,同时能达到较高的精度和效率。
The exploitation of numerical methods for hyperbolic conservation laws is one of the key research fields in computational mathematics since 1930's, a lot of effective and influential numerical methods have been put forward and the GDQ method was one important and efficient scheme among them. Besides simpleness and legibility, the GDQM is studied and applied widely today since it adds no demand and restriction on the grids.
In this paper, we discussed hyperbolic conservation laws, their exact solutions and the background of the discrete GDQ method. For the discontinuous arising in the process of computation about the compressible flows which governed by hyperbolic conservation laws, we pay more attention on the high-order accuracy and high resolution in the construction of adaptive discrete GDQMs and uniform schemes.
We analyzed the principle of the adaptive discrete GDQM and uniform GDQM elaborately, including grids-generation, spatial discretization and temporal discretization. The reconstruction of numerical fluxes on the whole field is the key step in spatial discretization because of its crucial impact on efficiency. In temporal discretization, we employed TVD Runge-Kutta sheme.
The two GDQMs are both of TVD property.
Numerical tests show that our methods could capture shocks and rarefaction wave efficiently and successfully in comparison with the graphs of some typical numerical examples.
本文所做的工作如下:
介绍了双曲型守恒律方程(组)及其解析解,阐述了离散GDQ方法产生的背景,针对一维双曲型守恒律方程(组)具有间断解(激波)的可压缩流问题,给出高精度、高分辨率的自适应离散GDQM和形式一致的离散GDQM,并详细分析了这两个格式的计算原理,给出了构造这两个格式的基本步骤,即网格剖分、空间离散、时间离散。其中空间离散的重构步是最关键的,它是在整个区间上离散并重构函数,这样处理使得计算效率有明显提高;时间离散采用经典方法——TVD型Runge-Kutta时间离散法。
这两种离散GDQ格式都是TVD的。
通过典型算例表明这两个数值方法都能很好地模拟间断,清晰地捕捉到激波、稀疏波,同时能达到较高的精度和效率。
The exploitation of numerical methods for hyperbolic conservation laws is one of the key research fields in computational mathematics since 1930's, a lot of effective and influential numerical methods have been put forward and the GDQ method was one important and efficient scheme among them. Besides simpleness and legibility, the GDQM is studied and applied widely today since it adds no demand and restriction on the grids.
In this paper, we discussed hyperbolic conservation laws, their exact solutions and the background of the discrete GDQ method. For the discontinuous arising in the process of computation about the compressible flows which governed by hyperbolic conservation laws, we pay more attention on the high-order accuracy and high resolution in the construction of adaptive discrete GDQMs and uniform schemes.
We analyzed the principle of the adaptive discrete GDQM and uniform GDQM elaborately, including grids-generation, spatial discretization and temporal discretization. The reconstruction of numerical fluxes on the whole field is the key step in spatial discretization because of its crucial impact on efficiency. In temporal discretization, we employed TVD Runge-Kutta sheme.
The two GDQMs are both of TVD property.
Numerical tests show that our methods could capture shocks and rarefaction wave efficiently and successfully in comparison with the graphs of some typical numerical examples.
引文
[1]Courant R,Friedrichs K O,Lewy H.On the partial difference equations of mathematical physics.IBM Journal,March,1967
[2]Godunov S K.A finite difference method for the numerical computational of discontinuout solution of the equation for fluid dynamics.Soy.Math.Sb,47:271-306,1959
[3]Lax D,Wendroff B.Systems of conservation laws.Communications on Pure and Applied Mathmatics,Vol.13,217-237,1960
[4]Murmann E M,Cole J D.Calculation of plane steady transonic flows.AIAA Journal,Vol.9,1971
[5]MacCorrnack R W.The effect of viscosity in hyper-velocity impact cratering.AIAA paper,69-354,1969
[6]Beam R M,Warming R F.An implicit finite difference algorithms for hyperbolic systems in conservation law.Journal of Computational Physics,No.22,87-109,1976
[7]Jameson A,Schmidt W,Turkel E.Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes.AIAA paper,81-1259,1981
[8]Steger J L,Warming R F.Flux vector splitting of the inviscid gas-dynamics equations with application to finite difference methods.Journal of Computational Physics,No.40,1981
[9]Van L B,Flux vector splitting for Enler equations.Lecture Notes in physics,Vol.170,Berlin,1982
[10]Roe P L.Approximate Riemann solvers,parameter vectors and difference schemes.Journal of Computational Physics,No.43,357-372,1981
[11]Liou M S,Sterren C J.A new flux splitting scheme.Journal of Computational Physics,No.107,23-29,1993
[12]Liou M S.Progress towards an improved CFD method:AUSM~+.AIAA paper,95-1701,1995
[13]Harten A.High resolution schemes for hyperbolic conservation laws.Journal of Computational Physics,No.49,357-393,1983
[14]Chakravarthy S R,Osher S.A new class of high accuracy TVD scheme for hyperbolic conservation laws,AlAA paper,85-373,1985
[15]Yee H C.Conservation of explicit and implicit symmetric TVD schemes and three applications,Journal of Computational Physics,No.68,151-179,1987
[16]张涵信。无波动、无自由参数耗散差分格式。空气动力学学报,Vol.6(2),145-155,1988
[17]Zhang H X,Zhuang F G.NND schemes and their applications to numerical simulation of two-and three-dimension flows.Advances in Applied Mechanics,Vol.29,193-256,1992
[18]张来平,张涵信。NND格式在非结构网格中的推广。力学学报,Vol.28(2),135-142,1996
[19]R.Bellman,B.Kashef and J.Casti,Differential quadrature:a technique for the rapid solution of nonlinear partial differential equations,J.Comput.Phys,10(1972),40-52
[20]C.Shu and B.Richard,Application of generalized differential quadrature to solve two_dimensional incompressible Navier_Stokes equations,Int.J.Numer,Meth.Fluids,15(1992),791-798
[21]C.Shu and Y.T.Chew,Application of multi-domain GDQ method to analysis of wave guides with rectangular Boundaries,J.Electromagnetic Waves and Applications,13(1999),223-224
[22]C.Shu,Generalized differential-integral quadrature and application to the simulation of incompressible viscous flows including parallel computation,Ph.D.Thesis,University of Glasgow,1991
[23]Chang Shu,Bryan E.Richards.,Application of Generalized Differential Quadrature To Solve Two-Dimensional Incompressible Navier-Stokes Equations.International Journal For Numerical Methods In Fluids,VOL.15,791-798(1992)
[24]Goodman J B,Leveque R J.,On the accuracy of stable schemes for 2D scalar conservation laws.Math Comput,1985,45:15-21.
[25]王承尧,王正华,杨晓辉,计算流体力学及其并行算法,国防科技大学出版社,2000
[26]Harten A.,The artificial compression method for computation of shocks and contact discontinuities Ⅲ.Self adjusting hybrid schemes.Math Comp 1977,32:363-389
[27]郑华盛,赵宁,成娟.(2004)一维高精度离散GDQ方法,计算数学 Vol.26 P293
[28]JianXian Qiu,Chi-Wang Shu.Finite Difference Weno Schemes With Lax-Wendroff-Type Time Discretizations.
[29]C-W.Shu,Total-variation-diminishing time discretizations,SIAM J.Sci.Statist,Comput.,9(1988),pp.1073-1084.
[30]郑华盛,赵宁(2005)双曲型守恒率的一种高精度TVD差分格式,计算物理 Vol.22 P13
[2]Godunov S K.A finite difference method for the numerical computational of discontinuout solution of the equation for fluid dynamics.Soy.Math.Sb,47:271-306,1959
[3]Lax D,Wendroff B.Systems of conservation laws.Communications on Pure and Applied Mathmatics,Vol.13,217-237,1960
[4]Murmann E M,Cole J D.Calculation of plane steady transonic flows.AIAA Journal,Vol.9,1971
[5]MacCorrnack R W.The effect of viscosity in hyper-velocity impact cratering.AIAA paper,69-354,1969
[6]Beam R M,Warming R F.An implicit finite difference algorithms for hyperbolic systems in conservation law.Journal of Computational Physics,No.22,87-109,1976
[7]Jameson A,Schmidt W,Turkel E.Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes.AIAA paper,81-1259,1981
[8]Steger J L,Warming R F.Flux vector splitting of the inviscid gas-dynamics equations with application to finite difference methods.Journal of Computational Physics,No.40,1981
[9]Van L B,Flux vector splitting for Enler equations.Lecture Notes in physics,Vol.170,Berlin,1982
[10]Roe P L.Approximate Riemann solvers,parameter vectors and difference schemes.Journal of Computational Physics,No.43,357-372,1981
[11]Liou M S,Sterren C J.A new flux splitting scheme.Journal of Computational Physics,No.107,23-29,1993
[12]Liou M S.Progress towards an improved CFD method:AUSM~+.AIAA paper,95-1701,1995
[13]Harten A.High resolution schemes for hyperbolic conservation laws.Journal of Computational Physics,No.49,357-393,1983
[14]Chakravarthy S R,Osher S.A new class of high accuracy TVD scheme for hyperbolic conservation laws,AlAA paper,85-373,1985
[15]Yee H C.Conservation of explicit and implicit symmetric TVD schemes and three applications,Journal of Computational Physics,No.68,151-179,1987
[16]张涵信。无波动、无自由参数耗散差分格式。空气动力学学报,Vol.6(2),145-155,1988
[17]Zhang H X,Zhuang F G.NND schemes and their applications to numerical simulation of two-and three-dimension flows.Advances in Applied Mechanics,Vol.29,193-256,1992
[18]张来平,张涵信。NND格式在非结构网格中的推广。力学学报,Vol.28(2),135-142,1996
[19]R.Bellman,B.Kashef and J.Casti,Differential quadrature:a technique for the rapid solution of nonlinear partial differential equations,J.Comput.Phys,10(1972),40-52
[20]C.Shu and B.Richard,Application of generalized differential quadrature to solve two_dimensional incompressible Navier_Stokes equations,Int.J.Numer,Meth.Fluids,15(1992),791-798
[21]C.Shu and Y.T.Chew,Application of multi-domain GDQ method to analysis of wave guides with rectangular Boundaries,J.Electromagnetic Waves and Applications,13(1999),223-224
[22]C.Shu,Generalized differential-integral quadrature and application to the simulation of incompressible viscous flows including parallel computation,Ph.D.Thesis,University of Glasgow,1991
[23]Chang Shu,Bryan E.Richards.,Application of Generalized Differential Quadrature To Solve Two-Dimensional Incompressible Navier-Stokes Equations.International Journal For Numerical Methods In Fluids,VOL.15,791-798(1992)
[24]Goodman J B,Leveque R J.,On the accuracy of stable schemes for 2D scalar conservation laws.Math Comput,1985,45:15-21.
[25]王承尧,王正华,杨晓辉,计算流体力学及其并行算法,国防科技大学出版社,2000
[26]Harten A.,The artificial compression method for computation of shocks and contact discontinuities Ⅲ.Self adjusting hybrid schemes.Math Comp 1977,32:363-389
[27]郑华盛,赵宁,成娟.(2004)一维高精度离散GDQ方法,计算数学 Vol.26 P293
[28]JianXian Qiu,Chi-Wang Shu.Finite Difference Weno Schemes With Lax-Wendroff-Type Time Discretizations.
[29]C-W.Shu,Total-variation-diminishing time discretizations,SIAM J.Sci.Statist,Comput.,9(1988),pp.1073-1084.
[30]郑华盛,赵宁(2005)双曲型守恒率的一种高精度TVD差分格式,计算物理 Vol.22 P13