模拟非线性浅水波的Boussinesq方程并行算法研究
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摘要
Boussinesq方程是用于描述非线性色散波在浅水中传播的方程,以Boussinesq方程为基础的数学模型可以用来模拟复杂地形上波浪的传播情况以及波浪与结构物之间的相互作用,它可以模拟波浪的折射、绕射、反射及浅化等现象,因此在工程中具有广泛的应用。随着Boussinesq方程的不断改进,Boussinesq方程已经能进行深水区和大面积场的计算,但是受到计算机存储空间及CPU的限制,使用单个处理器计算已经不能满足工程实际的需要,本文从这一实际情况出发,进行Boussinesq方程的并行程序设计,结合高性能并行计算机集群系统来进行Boussinesq方程水波数值模拟计算。
本文首先综述了Boussinesq方程在色散性、非线性等方面的改进,回顾了并行算法在Boussinesq方程中的应用,介绍了并行计算的分类和性能分析的方法。本文采用的方程是Madsen和Sch?ffer(1995)推导所得的方程,该方程具有很好的色散性,能适用于变水深、斜坡等复杂地形。对该方程在空间上采用了中心紧致差分格式,时间上采用中心差分格式,使得在差分点个数不变的情况下取得了更高的精度。方程的求解采用ADI方法,使得方程的求解具有很好的稳定性。
本文采用了MPI+Fortran的并行程序语言编写程序,程序具有很好的可移植性。对该方程采用了区域分解的并行算法进行并行程序设计,在不同方向计算时进行不同的区域划分,在进行模型并行化的过程中考虑了进程的负载平衡、死锁的避免、虚拟进程的设置等问题,进程间的数据交换通过消息的发送和接收来进行,经模型验证并行程序计算结果和串行程序的计算结果是完全吻合的,并在一定程度上缩短了计算时间。
Boussinesq equation is a nonlinear dispersion wave equation in shallow water. The numerical models based on Boussinesq equations could simulate the sitution of waves propagate on uneven bottom and the influence between waves and structure. It could simulate refraction, diffraction, reflection, shoaling etc. Therefore, Boussinesq equation models have been used widely. Along with continuous improvement of Boussinesq equation, it could simulate deep water region and great area field, however, computing only by one processor could not statisfy the projects’actual need because of the limitation of the memory’s size and CPU’s speed of computer. In this thesis, it designs parallel program of Boussinesq equation in order to solve this problem, and utilizes high performance parallel machine cluster system to Boussinesq equation’s wave simulation.
In this thesis, the improvements of dispersive property, shoaling, the application of parallel metods of Boussinesq equations are reviewed. Then Parallel machine classification and performance analysis method are introduced. It adopts the Boussinesq equation set up by Madsen and Sch?ffer(1995). This equation has better dispersive property than ever; it can be used on terrain of changed depth of water and slope. The center compact difference in space and center difference in time are applied in Boussinesq equation. As a result, it can obtain higher precision than the center difference in space while the number of difference points is not changed. ADI method is applied while solving this difference equation. It ensures that the solution process is stable.
MPI+Fortran parallel program language is used in this paper. The program has good portability. Adopting region decomposition method to design parallel program, the thesis uses different decomposition region at the different computation direction. It considers the load balance, avoiding dead lock, setting dummy processor and so on. The data’s exchange among processors uses“send and receive”method. At last, this parallel program is confirmed by models, the result of calculation tallies with the serial result. Paralle program reduces the computing time in a certain degree.
本文首先综述了Boussinesq方程在色散性、非线性等方面的改进,回顾了并行算法在Boussinesq方程中的应用,介绍了并行计算的分类和性能分析的方法。本文采用的方程是Madsen和Sch?ffer(1995)推导所得的方程,该方程具有很好的色散性,能适用于变水深、斜坡等复杂地形。对该方程在空间上采用了中心紧致差分格式,时间上采用中心差分格式,使得在差分点个数不变的情况下取得了更高的精度。方程的求解采用ADI方法,使得方程的求解具有很好的稳定性。
本文采用了MPI+Fortran的并行程序语言编写程序,程序具有很好的可移植性。对该方程采用了区域分解的并行算法进行并行程序设计,在不同方向计算时进行不同的区域划分,在进行模型并行化的过程中考虑了进程的负载平衡、死锁的避免、虚拟进程的设置等问题,进程间的数据交换通过消息的发送和接收来进行,经模型验证并行程序计算结果和串行程序的计算结果是完全吻合的,并在一定程度上缩短了计算时间。
Boussinesq equation is a nonlinear dispersion wave equation in shallow water. The numerical models based on Boussinesq equations could simulate the sitution of waves propagate on uneven bottom and the influence between waves and structure. It could simulate refraction, diffraction, reflection, shoaling etc. Therefore, Boussinesq equation models have been used widely. Along with continuous improvement of Boussinesq equation, it could simulate deep water region and great area field, however, computing only by one processor could not statisfy the projects’actual need because of the limitation of the memory’s size and CPU’s speed of computer. In this thesis, it designs parallel program of Boussinesq equation in order to solve this problem, and utilizes high performance parallel machine cluster system to Boussinesq equation’s wave simulation.
In this thesis, the improvements of dispersive property, shoaling, the application of parallel metods of Boussinesq equations are reviewed. Then Parallel machine classification and performance analysis method are introduced. It adopts the Boussinesq equation set up by Madsen and Sch?ffer(1995). This equation has better dispersive property than ever; it can be used on terrain of changed depth of water and slope. The center compact difference in space and center difference in time are applied in Boussinesq equation. As a result, it can obtain higher precision than the center difference in space while the number of difference points is not changed. ADI method is applied while solving this difference equation. It ensures that the solution process is stable.
MPI+Fortran parallel program language is used in this paper. The program has good portability. Adopting region decomposition method to design parallel program, the thesis uses different decomposition region at the different computation direction. It considers the load balance, avoiding dead lock, setting dummy processor and so on. The data’s exchange among processors uses“send and receive”method. At last, this parallel program is confirmed by models, the result of calculation tallies with the serial result. Paralle program reduces the computing time in a certain degree.
引文
[1] Witting, J.M., A numerical model for the evolution of non-linear water waves, J Comput Phys, 1984, 56: 203～236
[2] Madsen, P.A., Russel Murray, S?rensen, O.R., A new form of the Boussinesq equations with improved linear dispersion characteristics, Coastal Engineering, 1991, 15: 371～388
[3] Okey Nwogu, Alternative form of Boussinesq equations for nearshore wave propagation, Journal of Waterway, Port, Coastal and Ocean Engineering, 1993, 119(6): 618～638
[4] Sch?ffer, H.A., Madsen, P.A., Further enhancements of Boussinesq-type equations, Coastal Engineering, 1995, 26: 1～14
[5] Ge Wei, Kirby, J.T., Grilli, S.T. et al., A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves, J. Fluid Mech., 1995, 294: 71～92
[6] Gobbi, M.F., Kirby, J.T., A fourth order Boussinesq-type wave model, Coastal Engineering, 1996, 87: 1116～1129
[7] Madsen,P.A., Banijamali,B., Sch?ffer, H.A., et al., Boussinesq type equations with high accuracy in dispersion and nonlinear, Coastal Engineering, 1996, 8: 95～109
[8] Madsen, P.A., Binggham, H.B., Hua Liu, A new Boussinesq method for fully nonlinear waves from shallow to deep water, J. Fluid Mech. , 2002, 462: 1～30
[9] Khairil Irfan Sitanggang, Patrick Lynett, Parallel computation of a highly nonlinear Boussinesq equation model through domain decomposition, Int. J. Numer. Meth. Fluids, 2005, 49: 57～74
[10] Barry Wilkinson, Michael Allen,并行程序设计(陆鑫达译) ,北京:机械工业出版社, 2002, 2～3
[11]都志辉,李三立等,高性能计算之并行编程技术——MPI并行程序设计,北京:清华大学出版社, 2001, 1～5
[12]王韬,李晓明, SMP Cluster:如何开发两级并行,计算机工程与科学, 2002, 24: 78～81
[13]李晓梅,莫则尧等,可扩展并行算法的设计与分析,北京:国防工业出版社, 2000: 2～109
[14]龚卫华,金蓉,李跃新, PVM与MPI关于通性方法的比较,湖北大学学报(自然科学版), 2003, 25(2: 117～120
[15]孙晗琦,并行计算在计算流体力学中的研究(硕士学位论文),大连:大连理工大学: 2005
[16] Quinn, M.J., MPI与OpenMP并行程序设计(陈文光,武永卫等译),北京:清华大学出版社, 2004, 1～359
[17] Cai, X., Pedersen, G.K., Langtangen, H.P., A Parallel multi-subdomain strategy for solving Boussinesq water wave equations, Advances in Water Resources, 2005, 28: 215～233
[18] Lynett, P., Liu, P.L.-F., A two-layer approach to high order Boussinesq models., Journal of Fluid Mechanics, 1999, 399: 319～333.
[19] Wu, D.M., Wu, T.Y., Three-dimensional nonlinear long waves due to moving surface pressure, In Proceedings of the 14th Symposium on Naval Hydrodynamatics, MI, USA, 1982, 103～129
[20] Madsen, P.A., S?rensen, O.R., A new form of the Boussinesq equations with improved linear dispersion characteristics, Part2.A slowly-varying bathymetry, Coastal engineering, 1992, 18: 183～204
[21]钦文婷, Boussinesq方程数学模型的改进及其工程应用(硕士学位论文),天津:天津大学, 2003
[22]陶建华,夏中华,吴岩等,不规则波折射绕射综合数值模拟,天津大学力学系, 1994
[23]陶建华,水波的数值模拟,天津:天津大学出版社, 2005
[24] Abbott, M.B., McCowan, A.D., Warren, I.R., Accuracy of short wave numerical models, J. Hydr. Engrg, 1984, 110(10): 1287～1301
[25] Chawla, A., Kirby, J.T., A source function method for generation of waves on currents in Boussinesq models, Applied Ocean Research, 2000, 22: 75～83
[26] Wang, H.H., Parallel method for tridiagonal equations, ACM Transactions on Mathematical Software, 1981, 7(2): 170～183
[27] Mattor, Nathan, Williams, et al., Algorithm for solving tridiagonal matrix problems in parallel, Parallel Computing, 1995, 21(11): 1769～1782
[28] Peregrine, D.H., Long waves on a beach, Journal of Fluid Mechanics, 1967, 27(4): 815～827
[29] Ge Wei, Kirby, J.T., Grilli, S.T., et al., A fully nonlinear Boussinesq model for surface waves, Part 1. Highly nonlinear unsteady waves, Journal of Fluid Mechanics, 1995, 294: 71～92
[30] Kennedy, A.B., Kirby, J.T., Chen, Q., et al., Boussinesq-type equations with improved nonlinear performance, Wave Motion, 2001, 33(3): 225～243
[31] Lynett, P., Liu, P.L.-F., A numerical study of submarine-landslide-generate d waves and run-up, Proceedings of the Royal Society of London A, 2002, 458: 2885～2910
[32] Madsen, P.A. , Bingham, H.B., Liu, H., A new Boussinesq method for fully nonlinear waves from shallow to deep water, Journal of Fluid Mechanics, 2002, 462: 1～30
[33] Abbott, M.B., Petersen, H.M., Skovgaard, O., On the numerical modeling of short waves in shallow water, Journal of Hydraulic Research, 1978, 16(3): 173～204
[34] Bamberger, A. , Glowinski, R. , Tran, Q.H., A domain decomposition method for the acoustic wave equation with discontinuous coefficients and grid change, SIAM Journal on Numerical Analysis, 1997, 34(2): 603～639
[35] Benamou, J.-D., Desprès, B., A domain decomposition method for the Helmholtz equation and related optimal control problems, Journal of Computational Physics, 1997, 136(1): 68～82
[36] Bruaset, A.M., Cai, X., Langtangen, H.P., et al., Numerical solution of PDEs on parallel computers utilizing sequential simulators, Lect. Notes Comput. Sci., 1997, 161～168
[37] Cai, X., Overlapping domain decomposition methods, Advanced Topics in Computational Partial Differential Equations-Numerical Methods and Diffpack Programming, 2003, 57～95
[38] Cai, X., Acklam, E., Langtangen, H.P., et al., Parallel computing, Lect. Notes Computat. Sci. Eng., 2003, 1～55
[39] Chan, T.F., Mathew, T.P., Domain decomposition algorithms, Acta Numerica, 1994, 64～143
[40] Després, B., Domain decomposition method and the Helmholtz problem II, Second International Conference on Mathematical and Numerical Aspects of Wave Propagation, 1993, 197～206
[41] Dolean, V., Lanteri, S., A domain decomposition approach to finite volume solutions of the Euler equations on unstructured triangular meshes, International Journal for Numerical Methods in Fluids, 2001, 37(6): 625～656
[42] Langtangen, H.P., Pedersen, G.., Computational models for weakly dispersive nonlinear water waves, Computer Methods in Applied Mechanics and Engineering, 1998, 160(3-4): 337～358
[43] Woo, S.-B., Liu, P.L.-F., Finite-element model for modified Boussinesq equations. I: Model development, Journal of Waterway, Port, Coastal and Ocean Engineering, 2004, 130(1): 1～16
[44] Zelt, J.A., Raichlen, F., Lagrangian model for wave-induced harbour oscillations, Journal of Fluid Mechanics, 1990, 213: 203～225
[45] Agnon, Y., Madsen, P.A., Schaf?fer, H.A., A new approach to high-order Boussinesq models, Journal of Fluid Mechanics, 1999, 399: 319～333
[46] Madsen, P.A., Schaf?fer, H.A., Higher order Boussinesq-type equations for surface gravity waves-Derivation and analysis, Phil. Trans. R. Soc. Lond. A, 1998, 356: 1～59
[47] Madsen, P.A., Schaf?fer, H.A., A review of Boussinesq-type equations for gravity waves, Advances in Coastal and Ocean Engineering, 1999, 5: 1～95
[48] Madsen, P.A., Sorensen, O.R., Bound waves and triad interactions in shallow water, Ocean Engineering, 1993, 20(4): 359～388
[49] Rayleigh, L., On waves, Phil. Mag, 1876, 5(1): 257～279
[50] Schaf?fer, H.A., Second-order wavemaker theory for irregular waves, Ocean Engineering, 1996, 23(1): 47～88
[51] Wu, T.Y., Bidirectional solution street, Applied Mathematics and Mechanics (English Edition), 1995, 16(11): 289～306
[52] Wu, T.Y., Nonlinear waves and solitons in water, Physica D: Nonlinear Phenomena, 1998, 123(1～4): 48～63
[53] Wu, T.Y., Modeling nonlinear dispersive water waves, Journal of Engineering Mechanics, 1999, 125(7): 747～755
[54]李雪松,徐建中,一种非定常并行隐式算法的开发,超级计算通讯, 2006, 14(1): 25～28
[2] Madsen, P.A., Russel Murray, S?rensen, O.R., A new form of the Boussinesq equations with improved linear dispersion characteristics, Coastal Engineering, 1991, 15: 371～388
[3] Okey Nwogu, Alternative form of Boussinesq equations for nearshore wave propagation, Journal of Waterway, Port, Coastal and Ocean Engineering, 1993, 119(6): 618～638
[4] Sch?ffer, H.A., Madsen, P.A., Further enhancements of Boussinesq-type equations, Coastal Engineering, 1995, 26: 1～14
[5] Ge Wei, Kirby, J.T., Grilli, S.T. et al., A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves, J. Fluid Mech., 1995, 294: 71～92
[6] Gobbi, M.F., Kirby, J.T., A fourth order Boussinesq-type wave model, Coastal Engineering, 1996, 87: 1116～1129
[7] Madsen,P.A., Banijamali,B., Sch?ffer, H.A., et al., Boussinesq type equations with high accuracy in dispersion and nonlinear, Coastal Engineering, 1996, 8: 95～109
[8] Madsen, P.A., Binggham, H.B., Hua Liu, A new Boussinesq method for fully nonlinear waves from shallow to deep water, J. Fluid Mech. , 2002, 462: 1～30
[9] Khairil Irfan Sitanggang, Patrick Lynett, Parallel computation of a highly nonlinear Boussinesq equation model through domain decomposition, Int. J. Numer. Meth. Fluids, 2005, 49: 57～74
[10] Barry Wilkinson, Michael Allen,并行程序设计(陆鑫达译) ,北京:机械工业出版社, 2002, 2～3
[11]都志辉,李三立等,高性能计算之并行编程技术——MPI并行程序设计,北京:清华大学出版社, 2001, 1～5
[12]王韬,李晓明, SMP Cluster:如何开发两级并行,计算机工程与科学, 2002, 24: 78～81
[13]李晓梅,莫则尧等,可扩展并行算法的设计与分析,北京:国防工业出版社, 2000: 2～109
[14]龚卫华,金蓉,李跃新, PVM与MPI关于通性方法的比较,湖北大学学报(自然科学版), 2003, 25(2: 117～120
[15]孙晗琦,并行计算在计算流体力学中的研究(硕士学位论文),大连:大连理工大学: 2005
[16] Quinn, M.J., MPI与OpenMP并行程序设计(陈文光,武永卫等译),北京:清华大学出版社, 2004, 1～359
[17] Cai, X., Pedersen, G.K., Langtangen, H.P., A Parallel multi-subdomain strategy for solving Boussinesq water wave equations, Advances in Water Resources, 2005, 28: 215～233
[18] Lynett, P., Liu, P.L.-F., A two-layer approach to high order Boussinesq models., Journal of Fluid Mechanics, 1999, 399: 319～333.
[19] Wu, D.M., Wu, T.Y., Three-dimensional nonlinear long waves due to moving surface pressure, In Proceedings of the 14th Symposium on Naval Hydrodynamatics, MI, USA, 1982, 103～129
[20] Madsen, P.A., S?rensen, O.R., A new form of the Boussinesq equations with improved linear dispersion characteristics, Part2.A slowly-varying bathymetry, Coastal engineering, 1992, 18: 183～204
[21]钦文婷, Boussinesq方程数学模型的改进及其工程应用(硕士学位论文),天津:天津大学, 2003
[22]陶建华,夏中华,吴岩等,不规则波折射绕射综合数值模拟,天津大学力学系, 1994
[23]陶建华,水波的数值模拟,天津:天津大学出版社, 2005
[24] Abbott, M.B., McCowan, A.D., Warren, I.R., Accuracy of short wave numerical models, J. Hydr. Engrg, 1984, 110(10): 1287～1301
[25] Chawla, A., Kirby, J.T., A source function method for generation of waves on currents in Boussinesq models, Applied Ocean Research, 2000, 22: 75～83
[26] Wang, H.H., Parallel method for tridiagonal equations, ACM Transactions on Mathematical Software, 1981, 7(2): 170～183
[27] Mattor, Nathan, Williams, et al., Algorithm for solving tridiagonal matrix problems in parallel, Parallel Computing, 1995, 21(11): 1769～1782
[28] Peregrine, D.H., Long waves on a beach, Journal of Fluid Mechanics, 1967, 27(4): 815～827
[29] Ge Wei, Kirby, J.T., Grilli, S.T., et al., A fully nonlinear Boussinesq model for surface waves, Part 1. Highly nonlinear unsteady waves, Journal of Fluid Mechanics, 1995, 294: 71～92
[30] Kennedy, A.B., Kirby, J.T., Chen, Q., et al., Boussinesq-type equations with improved nonlinear performance, Wave Motion, 2001, 33(3): 225～243
[31] Lynett, P., Liu, P.L.-F., A numerical study of submarine-landslide-generate d waves and run-up, Proceedings of the Royal Society of London A, 2002, 458: 2885～2910
[32] Madsen, P.A. , Bingham, H.B., Liu, H., A new Boussinesq method for fully nonlinear waves from shallow to deep water, Journal of Fluid Mechanics, 2002, 462: 1～30
[33] Abbott, M.B., Petersen, H.M., Skovgaard, O., On the numerical modeling of short waves in shallow water, Journal of Hydraulic Research, 1978, 16(3): 173～204
[34] Bamberger, A. , Glowinski, R. , Tran, Q.H., A domain decomposition method for the acoustic wave equation with discontinuous coefficients and grid change, SIAM Journal on Numerical Analysis, 1997, 34(2): 603～639
[35] Benamou, J.-D., Desprès, B., A domain decomposition method for the Helmholtz equation and related optimal control problems, Journal of Computational Physics, 1997, 136(1): 68～82
[36] Bruaset, A.M., Cai, X., Langtangen, H.P., et al., Numerical solution of PDEs on parallel computers utilizing sequential simulators, Lect. Notes Comput. Sci., 1997, 161～168
[37] Cai, X., Overlapping domain decomposition methods, Advanced Topics in Computational Partial Differential Equations-Numerical Methods and Diffpack Programming, 2003, 57～95
[38] Cai, X., Acklam, E., Langtangen, H.P., et al., Parallel computing, Lect. Notes Computat. Sci. Eng., 2003, 1～55
[39] Chan, T.F., Mathew, T.P., Domain decomposition algorithms, Acta Numerica, 1994, 64～143
[40] Després, B., Domain decomposition method and the Helmholtz problem II, Second International Conference on Mathematical and Numerical Aspects of Wave Propagation, 1993, 197～206
[41] Dolean, V., Lanteri, S., A domain decomposition approach to finite volume solutions of the Euler equations on unstructured triangular meshes, International Journal for Numerical Methods in Fluids, 2001, 37(6): 625～656
[42] Langtangen, H.P., Pedersen, G.., Computational models for weakly dispersive nonlinear water waves, Computer Methods in Applied Mechanics and Engineering, 1998, 160(3-4): 337～358
[43] Woo, S.-B., Liu, P.L.-F., Finite-element model for modified Boussinesq equations. I: Model development, Journal of Waterway, Port, Coastal and Ocean Engineering, 2004, 130(1): 1～16
[44] Zelt, J.A., Raichlen, F., Lagrangian model for wave-induced harbour oscillations, Journal of Fluid Mechanics, 1990, 213: 203～225
[45] Agnon, Y., Madsen, P.A., Schaf?fer, H.A., A new approach to high-order Boussinesq models, Journal of Fluid Mechanics, 1999, 399: 319～333
[46] Madsen, P.A., Schaf?fer, H.A., Higher order Boussinesq-type equations for surface gravity waves-Derivation and analysis, Phil. Trans. R. Soc. Lond. A, 1998, 356: 1～59
[47] Madsen, P.A., Schaf?fer, H.A., A review of Boussinesq-type equations for gravity waves, Advances in Coastal and Ocean Engineering, 1999, 5: 1～95
[48] Madsen, P.A., Sorensen, O.R., Bound waves and triad interactions in shallow water, Ocean Engineering, 1993, 20(4): 359～388
[49] Rayleigh, L., On waves, Phil. Mag, 1876, 5(1): 257～279
[50] Schaf?fer, H.A., Second-order wavemaker theory for irregular waves, Ocean Engineering, 1996, 23(1): 47～88
[51] Wu, T.Y., Bidirectional solution street, Applied Mathematics and Mechanics (English Edition), 1995, 16(11): 289～306
[52] Wu, T.Y., Nonlinear waves and solitons in water, Physica D: Nonlinear Phenomena, 1998, 123(1～4): 48～63
[53] Wu, T.Y., Modeling nonlinear dispersive water waves, Journal of Engineering Mechanics, 1999, 125(7): 747～755
[54]李雪松,徐建中,一种非定常并行隐式算法的开发,超级计算通讯, 2006, 14(1): 25～28