高精度差分格式数学特性分析
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摘要
为了在众多的差分格式中筛选出具有较高的计算精度同时兼顾较高计算效率的差分格式,本文对一些著名的格式进行了比较分析。首先介绍了模型方程的一个可允许弱解,利用经典格式、一阶TVD格式、Harten格式、Godunov格式、二阶Godunov格式(MUSCL方法)和三阶Godunov格式(PPM方法)对模型方程进行求解,又介绍了欧拉方程Riemann问题解类型的判断规则及求解方法,用Roe格式、Harten格式、AUSM格式求解欧拉方程激波管问题,最后用Crank-Nicolson半隐格式和高精度全隐紧致格式求解线性Burgers方程。利用Fortran95中的cpu-time函数在单任务状态下计算上述差分格式的计算时间,分析计算效率。经过算例检验证实:在对模型方程的求解中,三阶Godunov格式(PPM方法)具有较高的计算精度及计算效率。在对欧拉方程的求解中,AUSM格式虽然具有计算速度快的特点,但仅具有一阶精度。在对线性Burgers方程的求解中,高精度全隐紧致格式在空间上达到四阶精度,同时又保持了较高的计算效率。
For the sake of seeking an scheme with both accuracy and efficiency, this article compare and analyze some famous schemes. Firstly, this article introduces a permissible weak solution, use classical schemes、first order TVD scheme、Harten scheme、Godunov scheme、second order Godunov scheme(MUSCL method) and third order Godunov scheme(PPM method) to solve model equation. Secondly, this article introduces the regulation of how to judge the type of Riemann solution and the method of how to solve, use Roe scheme、Harten scheme and AUSM scheme to solve the shock tube problem of Euler equation. Finally, this article use Crank-Nicolson scheme、High-Order compact implicit difference method to solve linear Burgers equation and use the cpu-time function provided by Fortran95 to test the efficiency of above schemes. Numerical results prove: When solve the model equation, third order Godunov scheme (PPM method) is more accurate and efficient than other schemes, When solve the Euler equation, although AUSM scheme’s computational speed is fast, the scheme only have first order accuracy, When solve the linear Burgers equation, High-Order compact implicit difference method have forth order in space and high efficiency.
For the sake of seeking an scheme with both accuracy and efficiency, this article compare and analyze some famous schemes. Firstly, this article introduces a permissible weak solution, use classical schemes、first order TVD scheme、Harten scheme、Godunov scheme、second order Godunov scheme(MUSCL method) and third order Godunov scheme(PPM method) to solve model equation. Secondly, this article introduces the regulation of how to judge the type of Riemann solution and the method of how to solve, use Roe scheme、Harten scheme and AUSM scheme to solve the shock tube problem of Euler equation. Finally, this article use Crank-Nicolson scheme、High-Order compact implicit difference method to solve linear Burgers equation and use the cpu-time function provided by Fortran95 to test the efficiency of above schemes. Numerical results prove: When solve the model equation, third order Godunov scheme (PPM method) is more accurate and efficient than other schemes, When solve the Euler equation, although AUSM scheme’s computational speed is fast, the scheme only have first order accuracy, When solve the linear Burgers equation, High-Order compact implicit difference method have forth order in space and high efficiency.
引文
1杨志峰,王煊.高精度有限差分方法研究.自然科学进展,第九卷,第九期,1999
2 Strikwerd J C. High-order-accurate schemes for incompressible viscous flow. Int J Numer Methods Fluids,1997,24:715～734
3 Thom A, Apelt C J. Field Computations in Engineering and Physics. London: C Van Nostrand Company Ltd, 1961
4 Roberts K V, Weiss N O. Convective difference schemes. Mathematics of Computation, 1966,20(94):272～299
5 Rusanov V V . On difference schemes of third order accuracy for nonlinear hyperbolic systems. J Comput Phys, 1970,5:507～516
6 Burstein S Z, Mirin A A .Third order difference methods for hyperbolic equations. J Comput Phy, 1970,5:547～571
7 Warming R F, Kutler P, Lomax H.Second-and third-order noncentered difference schemes for nonlinear hyperbolic equations. AIAAJ,1973,11:189～196
8 CHA G C and BILGEN E. Numerical solutions of Euler equations by using a new flux vector splitting scheme[J ] . International J . for Numerical Methods in Fluids , 1993 ,17 ,1152144.
9 EDWARDS J R. A low2diffusion flux2splitting scheme for NS calculations [R] . AIAA 9521703.
10 HANEL D , SCHWANE R. An implicit flux2vector splitting scheme for the computation of viscous hypersonic flow [R] . AIAA 8920274.
11 LIOU M-S,STERREN C J . A new flux splitting scheme [J ] . Journal of computational physics 107, 23～39, 1993
12 LIOU M-S, A Sequel to AUSM: AUSM +. Journal of computational physicsphysics 129, 364～382(1996)
13 LIOU M-S, A sequel to AUSM, part II : AUSM +-up for all speeds. Journal of computational physicsphysics 214, 137～170(2006)
14 Chih-Hao Chang , M-S Liou. A robust and accurate approach to computing compressible multiphase flow: Stratified flow model and AUSM+-up scheme. Journal of Computational Physics. In Press, Corrected Proof, Available online 192007
15 J.R. García-Cascales, H. Paillère. Application of AUSM schemes to multi-dimensional compressible two-phase flow problems. Volume 236, Issue 12, June 2006, Pages 1225-1239
16 D. Vigneron, J.-M. Vaassen , J.-A. Essers. An implicit finite volume method for the solution of 3D low Mach number viscous flows using a local preconditioning technique. Journal of Computational and Applied Mathematics, In Press, Corrected Proof, Available online 19 2006
17梁德望,王可. AUSM +格式的改进.空气动力学报.第22卷,第4期, 2004
18孙迎丹,王刚. AUSM +-up格式在无网格算法中的推广.空气动力学报.第23卷,第4期, 2005
19赵广慧,陶建华.浅水非线形Boussinesq方程的紧致差分格式研究.第八届全国计算流体力学会议论文集.北京:中国力学学会,1996.480～485
20 Holly F M, Usseglio-Polatera J M. Dispersion simulation in two-dimensional tidal flow. J Hydrau Engng,1984,110(7):905～926
21 Dennis S C R, Hundson J D. Compact h 4 finite difference approximations to operators of Navier-Stokes type. J Comput Phys, 1989,85(2):390～416
22 Nobuhiro B, Hideaki M, Hisashi K. Compact finite difference schemes foe the flow around cylinder. J of the Society of Naval Architects of Japan, 1987, 159(33):25～40
23 Lele S K. Compact finite difference schemes with spectral-like resolution. J Comput Phys, 1992, 103:16～42
24 Fu D X, Ma Y W. On efficiency and accuracy of numerical methods for solving the aerodynamics equations. In Proceedings of Third international Symposium on CFD, Nayoya: Japan Society of CFD,1989
25 Fu D X, Ma Y W. High resolution schemes. In: Hafez M, Oshima K,eds. Computational Fluid Dynamics Review 1995.Chichester: John Wiley & Sons Ltd, 1995.234～249
26 Ma Y W, Fu D X. Finite difference schemes with super high order of accuracy. In: Proceeding of ASIA Workshop on CFD, Mianyang, 1994
27张健.计算流体力学在航空叶轮机设计中的应用.航空科学技术1999,4
28姜礼尚,陈亚浙,刘西垣,易法槐.数学物理方程讲义(第二版),高等教育出版社.
29张涵信.无波动、无自由参数的耗散差分格式.空气动力学报. 1988 6(2):143
30水鸿寿.一维流体力学差分方法.北京:国防工业出版社.1998
31 A. Harten, A high resolution scheme for the computation of weak solutions of hyperbolid conservation laws, J. Compt .Phys. 49,357(1983)
32 Yee H C, Warming R F, Harten A. Implicit total variation Diminishing (TVD) schemes for steady-state calculations.AIAA 83-1902
33 Godunov C K.Difference method for computing the discontinuity in fluid dynamics.Math Sbornik,1959,47(3):271
34 Godunov S K et al. Numerical Solution of Multidimensional Problems in Gas Dynamics, Nauka, Moscow, 1976 (in Russian).
35 Van Leer B.Towards the ultimate conservative scheme V.A second-order sequel to Godunov’s method .J. Comput. Phys,1979,32:101～136
36 Colella P, and Wood word. The Piecewise Parabolic Method (PPM) for Gas-Dynamical Simulations, J. Comput. Phys., 1984,54:174～201.
37 Eidelman S, Colella P, and Shreeve R P. Application of the Godunov Method and its Second-Order Extension to Cascade Flow Modeling, AIAA J., 1984, 22(1):1609～1615.
38 J. Adams, W. Brainerd, J. Martin, and J. Wagener. Fortran95 Handbook
39 Roe P L. Approximate Reimann solver, parameter vectors and difference scheme. J. Compt .Phys, 1981, 43: 357～372
40付德薰,马延文.计算流体力学.北京:高等教育出版社,2002
41涂国华,袁湘江,夏治强,呼振.一类TVD型的迎风紧致差分格式.应用数学与力学.第27卷,第6期,2006.
42 Ratnesh K. Shukla, Mahidhar Tatineni, Xiaolin Zhong. Very high-order compact finite difference schemes on non-uniform grids for incompressible Navier–Stokes equations. ournal of Computational Physics, Volume 224, Issue 2, 10 June 2007, Pages 1064-1094
43 Godehard Sutmann. Compact finite difference schemes of sixth order for the Helmholtz equation. Journal of Computational and Applied Mathematics, Volume 203, Issue 1, 1 June 2007, Pages 15-31.
44 Ming-Chih Lai , Jui-Ming Tseng. A formally fourth-order accurate compact scheme for 3D Poisson equation in cylindrical coordinates. Journal ofComputational and Applied Mathematics, Volume 201, Issue 1, 1 April 2007, Pages 175-181.
45 Sergio Pirozzoli. Performance analysis and optimization of finite-difference schemes for wave propagation problems. Journal of Computational Physics, Volume 222, Issue 2, 20 March 2007, Pages 809-831.
46 Tamar Eilam, Cyril Gavoille, David Peleg. Average stretch analysis of compact routing schemes. Discrete Applied Mathematics, Volume 155, Issue 5, 15 March 2007, Pages 598-610.
47 Z.F. Tian, Y.B. Ge. A fourth-order compact ADI method for solving two-dimensional unsteady convection–diffusion problems. Journal of Computational and Applied Mathematics, Volume 198, Issue 1, 1 January 2007, Pages 268-286.
48葛永斌,田振夫,吴文权.含源项非定常对流扩散方程的高精度紧致隐式差分法.水动力学研究与进展.A辑第21卷第5期.2006
49 STRIKWERDA J C. Finite Difference Schemes & Partial Differential Equations [M]. New York Champan & Hall, 1989
2 Strikwerd J C. High-order-accurate schemes for incompressible viscous flow. Int J Numer Methods Fluids,1997,24:715～734
3 Thom A, Apelt C J. Field Computations in Engineering and Physics. London: C Van Nostrand Company Ltd, 1961
4 Roberts K V, Weiss N O. Convective difference schemes. Mathematics of Computation, 1966,20(94):272～299
5 Rusanov V V . On difference schemes of third order accuracy for nonlinear hyperbolic systems. J Comput Phys, 1970,5:507～516
6 Burstein S Z, Mirin A A .Third order difference methods for hyperbolic equations. J Comput Phy, 1970,5:547～571
7 Warming R F, Kutler P, Lomax H.Second-and third-order noncentered difference schemes for nonlinear hyperbolic equations. AIAAJ,1973,11:189～196
8 CHA G C and BILGEN E. Numerical solutions of Euler equations by using a new flux vector splitting scheme[J ] . International J . for Numerical Methods in Fluids , 1993 ,17 ,1152144.
9 EDWARDS J R. A low2diffusion flux2splitting scheme for NS calculations [R] . AIAA 9521703.
10 HANEL D , SCHWANE R. An implicit flux2vector splitting scheme for the computation of viscous hypersonic flow [R] . AIAA 8920274.
11 LIOU M-S,STERREN C J . A new flux splitting scheme [J ] . Journal of computational physics 107, 23～39, 1993
12 LIOU M-S, A Sequel to AUSM: AUSM +. Journal of computational physicsphysics 129, 364～382(1996)
13 LIOU M-S, A sequel to AUSM, part II : AUSM +-up for all speeds. Journal of computational physicsphysics 214, 137～170(2006)
14 Chih-Hao Chang , M-S Liou. A robust and accurate approach to computing compressible multiphase flow: Stratified flow model and AUSM+-up scheme. Journal of Computational Physics. In Press, Corrected Proof, Available online 192007
15 J.R. García-Cascales, H. Paillère. Application of AUSM schemes to multi-dimensional compressible two-phase flow problems. Volume 236, Issue 12, June 2006, Pages 1225-1239
16 D. Vigneron, J.-M. Vaassen , J.-A. Essers. An implicit finite volume method for the solution of 3D low Mach number viscous flows using a local preconditioning technique. Journal of Computational and Applied Mathematics, In Press, Corrected Proof, Available online 19 2006
17梁德望,王可. AUSM +格式的改进.空气动力学报.第22卷,第4期, 2004
18孙迎丹,王刚. AUSM +-up格式在无网格算法中的推广.空气动力学报.第23卷,第4期, 2005
19赵广慧,陶建华.浅水非线形Boussinesq方程的紧致差分格式研究.第八届全国计算流体力学会议论文集.北京:中国力学学会,1996.480～485
20 Holly F M, Usseglio-Polatera J M. Dispersion simulation in two-dimensional tidal flow. J Hydrau Engng,1984,110(7):905～926
21 Dennis S C R, Hundson J D. Compact h 4 finite difference approximations to operators of Navier-Stokes type. J Comput Phys, 1989,85(2):390～416
22 Nobuhiro B, Hideaki M, Hisashi K. Compact finite difference schemes foe the flow around cylinder. J of the Society of Naval Architects of Japan, 1987, 159(33):25～40
23 Lele S K. Compact finite difference schemes with spectral-like resolution. J Comput Phys, 1992, 103:16～42
24 Fu D X, Ma Y W. On efficiency and accuracy of numerical methods for solving the aerodynamics equations. In Proceedings of Third international Symposium on CFD, Nayoya: Japan Society of CFD,1989
25 Fu D X, Ma Y W. High resolution schemes. In: Hafez M, Oshima K,eds. Computational Fluid Dynamics Review 1995.Chichester: John Wiley & Sons Ltd, 1995.234～249
26 Ma Y W, Fu D X. Finite difference schemes with super high order of accuracy. In: Proceeding of ASIA Workshop on CFD, Mianyang, 1994
27张健.计算流体力学在航空叶轮机设计中的应用.航空科学技术1999,4
28姜礼尚,陈亚浙,刘西垣,易法槐.数学物理方程讲义(第二版),高等教育出版社.
29张涵信.无波动、无自由参数的耗散差分格式.空气动力学报. 1988 6(2):143
30水鸿寿.一维流体力学差分方法.北京:国防工业出版社.1998
31 A. Harten, A high resolution scheme for the computation of weak solutions of hyperbolid conservation laws, J. Compt .Phys. 49,357(1983)
32 Yee H C, Warming R F, Harten A. Implicit total variation Diminishing (TVD) schemes for steady-state calculations.AIAA 83-1902
33 Godunov C K.Difference method for computing the discontinuity in fluid dynamics.Math Sbornik,1959,47(3):271
34 Godunov S K et al. Numerical Solution of Multidimensional Problems in Gas Dynamics, Nauka, Moscow, 1976 (in Russian).
35 Van Leer B.Towards the ultimate conservative scheme V.A second-order sequel to Godunov’s method .J. Comput. Phys,1979,32:101～136
36 Colella P, and Wood word. The Piecewise Parabolic Method (PPM) for Gas-Dynamical Simulations, J. Comput. Phys., 1984,54:174～201.
37 Eidelman S, Colella P, and Shreeve R P. Application of the Godunov Method and its Second-Order Extension to Cascade Flow Modeling, AIAA J., 1984, 22(1):1609～1615.
38 J. Adams, W. Brainerd, J. Martin, and J. Wagener. Fortran95 Handbook
39 Roe P L. Approximate Reimann solver, parameter vectors and difference scheme. J. Compt .Phys, 1981, 43: 357～372
40付德薰,马延文.计算流体力学.北京:高等教育出版社,2002
41涂国华,袁湘江,夏治强,呼振.一类TVD型的迎风紧致差分格式.应用数学与力学.第27卷,第6期,2006.
42 Ratnesh K. Shukla, Mahidhar Tatineni, Xiaolin Zhong. Very high-order compact finite difference schemes on non-uniform grids for incompressible Navier–Stokes equations. ournal of Computational Physics, Volume 224, Issue 2, 10 June 2007, Pages 1064-1094
43 Godehard Sutmann. Compact finite difference schemes of sixth order for the Helmholtz equation. Journal of Computational and Applied Mathematics, Volume 203, Issue 1, 1 June 2007, Pages 15-31.
44 Ming-Chih Lai , Jui-Ming Tseng. A formally fourth-order accurate compact scheme for 3D Poisson equation in cylindrical coordinates. Journal ofComputational and Applied Mathematics, Volume 201, Issue 1, 1 April 2007, Pages 175-181.
45 Sergio Pirozzoli. Performance analysis and optimization of finite-difference schemes for wave propagation problems. Journal of Computational Physics, Volume 222, Issue 2, 20 March 2007, Pages 809-831.
46 Tamar Eilam, Cyril Gavoille, David Peleg. Average stretch analysis of compact routing schemes. Discrete Applied Mathematics, Volume 155, Issue 5, 15 March 2007, Pages 598-610.
47 Z.F. Tian, Y.B. Ge. A fourth-order compact ADI method for solving two-dimensional unsteady convection–diffusion problems. Journal of Computational and Applied Mathematics, Volume 198, Issue 1, 1 January 2007, Pages 268-286.
48葛永斌,田振夫,吴文权.含源项非定常对流扩散方程的高精度紧致隐式差分法.水动力学研究与进展.A辑第21卷第5期.2006
49 STRIKWERDA J C. Finite Difference Schemes & Partial Differential Equations [M]. New York Champan & Hall, 1989