一类非线性偏微分方程的若干有限差分格式
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摘要
因为很多物理现象和过程的数学模型都可以用非线性偏微分方程来表示,而这些非线性偏微分方程在很多情况下,求解精确解比较困难,故非线性偏微分方程的数值解法在数值分析中占有重要的地位。近几十年来,它的理论和方法都有了很大的发展,而且在各个科学技术领域中的应用也越来越广泛。
有限差分方法(FDM)作为一种数值离散方法,以其求解问题时的易操作性和较大的灵活性,在科学研究和工作计算中得到了广泛的应用。随着计算机技术的迅猛发展,如何精细准确地进行数值模拟已经成为重要课题。到目前为止,虽然出现了很多的有限差分方法,但是泰勒展开法作为一种比较经典,简单的方法,仍然在有限差分方程的数值求解中占有十分重要的地位。论文主要是应用泰勒展开法进行求解。
全文共分为五章。第一章介绍了国内外有限差分方法的发展史及现状,并给出了论文研究的主要结果和创新点。第二章给出了论文所需要的一些预备知识。第三、四、五章分别给出了三个方程的有限差分格式,并且对每一种差分格式的收敛性和稳定性给出了证明,其中第四章、第五章给出了相应的数值例子,通过数值例子可以更加清楚地看到,所给出的格式的有效性和可行性。
Because many mathematical models of the physical phenomenon and process all can be denoted by the nonlinear partial differential equations. In many causes, it is very difficult to solve the exact solution of these nonlinear PDES. Therefore, the numerical method of the PDES plays an important role in the numerical analysis. In these recent dozens of years, its theory and method have great development. Moreover, the application of its is more and more extensive in every technology field.
As a numerical discreatization method, for its easy operation and great agility character in the solving problem, the finite difference has gotten extensive application in the science research and computation. With the rapid development of the computer, how to get the fine and true numerical simulation has become an important task. Untill now, there are a lot of finite difference methods, but Taylor expandness method is very important in the finite difference methods as a classical and simply method. This paper has proposed by Taylor expandness method.
The full text altogether divides into five chapters. In first chapter, we introduce the finite difference theory history, the present situation and provide the typical study method and main result with innovation. The second chapter, some readiness knowledge is given. The third, fourth and fifth chapters, three finite difference schemes were proposed. Convergence and stability of the difference solution were proved. The fourth and fifth chapers, numerical results demonstrate that the methods proposing are efficient and reliable.
有限差分方法(FDM)作为一种数值离散方法,以其求解问题时的易操作性和较大的灵活性,在科学研究和工作计算中得到了广泛的应用。随着计算机技术的迅猛发展,如何精细准确地进行数值模拟已经成为重要课题。到目前为止,虽然出现了很多的有限差分方法,但是泰勒展开法作为一种比较经典,简单的方法,仍然在有限差分方程的数值求解中占有十分重要的地位。论文主要是应用泰勒展开法进行求解。
全文共分为五章。第一章介绍了国内外有限差分方法的发展史及现状,并给出了论文研究的主要结果和创新点。第二章给出了论文所需要的一些预备知识。第三、四、五章分别给出了三个方程的有限差分格式,并且对每一种差分格式的收敛性和稳定性给出了证明,其中第四章、第五章给出了相应的数值例子,通过数值例子可以更加清楚地看到,所给出的格式的有效性和可行性。
Because many mathematical models of the physical phenomenon and process all can be denoted by the nonlinear partial differential equations. In many causes, it is very difficult to solve the exact solution of these nonlinear PDES. Therefore, the numerical method of the PDES plays an important role in the numerical analysis. In these recent dozens of years, its theory and method have great development. Moreover, the application of its is more and more extensive in every technology field.
As a numerical discreatization method, for its easy operation and great agility character in the solving problem, the finite difference has gotten extensive application in the science research and computation. With the rapid development of the computer, how to get the fine and true numerical simulation has become an important task. Untill now, there are a lot of finite difference methods, but Taylor expandness method is very important in the finite difference methods as a classical and simply method. This paper has proposed by Taylor expandness method.
The full text altogether divides into five chapters. In first chapter, we introduce the finite difference theory history, the present situation and provide the typical study method and main result with innovation. The second chapter, some readiness knowledge is given. The third, fourth and fifth chapters, three finite difference schemes were proposed. Convergence and stability of the difference solution were proved. The fourth and fifth chapers, numerical results demonstrate that the methods proposing are efficient and reliable.
引文
1房少梅,吴荣俏.一类广义的Kdv方程的显式差分解.应用数学学报,2007,30 (2):368-376
2 L. Zhang. A Finite Difference Scheme for Generalized Regularized Long-wave Equation. Appl. Math. Comput,2005,168:962-972
3李荣华.偏微分方程数值解法.高等教育出版社,2006:13-183
4陆金甫,关治.偏微分方程数值解法.清华大学出版社,2005:45-55
5张文生.科学计算中的偏微分方程有限差分法.高等教育出版社,2006:275-282
6 V. V. Rusanov. On Difference Schemes of Third Order Accuracy for Nonlinear Hyperbolic Systems. J Comput Phys,1970,5:507-516
7 S. Z. Burstein, A. A. Mirin. Third Order Difference Methods for Hyperbolic Equations. J Comput Phys,1970,5:547-571
8 R. F. Warming, P. Kutler, H. Lomax. Second and Third-order Noncentered Difference Schemes for Nonlinear Hyperbolic Equations. AIAAJ,1973,11:189-196
9赵广慧,陶建华.浅水非线性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
10 S. K. Lele. Compact Finite Difference Schemes with Spectral-like Resolution. J Comput Phys,1992,103:16-42
11 B. Cockbum, C. W. Shu. Nonlinearly Stable Compact Schemes for Shock Caculations. SIAM J Numer Anal,1994,31(3):607-627
12 X. G. Deng, M. Hiroshi. Compact High-order Accurate Nonlinear Schemes. J Comput Phys,1997,130(1):77-91
13杨志峰,陈国谦.非定常对流扩散方程的无条件稳定高精度紧致差分格式.自然科学进展,1993,3(3):263-266
14杨志峰,周同明.含源扩散方程的一种高精度差分方法.北京师范大学学报(自然科学版),1992,28(1):124-128
15杨志峰,陈国谦.含源扩散方程的一种高精度差分方法(II).北京师范大学学报(自然科学版),1992,28(3):315-316
16杨志峰,许协庆.泊松方程的高精度优化差分方法.水动力学研究与进展,1992,(A) 7(3):263-269
17杨志峰,陈国谦,罗朝俊.一类高精度待定差分格式.见:王光谦,曲政,龚旗煌编,中国博士后论文集.北京:北京大学出版社,1991,(4):99-105
18杨志峰,陈国谦.含源汇非定常对流扩散问题紧致四阶差分格式.科学通报,1993, 38(2):113-116
19杨志峰.流体动力与传热的高级有限差分方法.见:国家自然科学青年基金研究成果报告会摘要汇编(力学).北京:科学出版社,1995:81-82
20陈国谦,杨志峰.Navier-Stokes方程的倒数型四阶紧致差分格式.见:王光谦,曲政,龚旗煌编,中国博士后论文集.北京:北京大学出版社,1991(4):132-141
21陈国谦,杨志峰,高智.对流扩散方程的四阶指数型差分格式.计算物理,1991,8(4): 359-372
22陈国谦,杨志峰.对流扩散方程的指数型摄动差分法.计算物理,1993,10(2):197-207
23 G. Q. Chen, Z. Gao, Z. F. Yang. A Pertubational h 4 Exponential Finite Difference Scheme for the Convective Diffusion Equation. J Comput Phys,1993,104(1):129-139
24 G. Q. Chen, Z. F. Yang. A Peturbational Fourth-order Upwind Finite Difference Scheme for the Convection-diffusion Equation. Journal of Hydrodynamics,1993,(B), 5(1): 82-97
25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
26 R. S. Hirsh. Higher Order Accurate Difference Solutions of Fluid Mechanics Problems by a Compact Differencing Technique. J Comput Phys,1975,19(1):90-109
27 I. Dag, D. Irk, B. Saka. A Numerical Solution of the Burger’s Equation Using Cubic B-splines. Applied Mathematics and Computation,2005,163:199-211
28 B. A. Refik, M. A. Saglam. A Mixed Finite Difference and Boundary Element Approach to One-dimensional Burgers’Equation. Applied Mathematics and Computation,2005, 160:663-673
29 E. M. Ronald. A Nonstandard Finite Difference Scheme for the Diffusionless BurgersEquation with Logistic Reaction. Mathematics and Computers in Simulation, 2003,62:117-124
30 M. A. Hela, M. S. Mehanna. A Comparison between Two Different Methods for Solving Kdv-Burgers Equation. Chaos, Solitons and Fractals,2006,28:320-326
31彭亚绵,闵涛,张世梅等.Burgers方程的MOL数值解法.西安理工大学学报,2004,20 (3):276-279
32 H. A. Amir, E. Shakour, M. Dehghan. Some Implicit Methods for the Numerical Solution of Burgers’Equation. Applied Mathematics and Computation,2007, 191:560-570
33 K. K. Mohan, A. Awasthi. A Numerical Method Based on Crank-Nicolson Scheme for Burgers’Equation. Applied Mathematics and Computation,2006,182:1430-1442
34 A. Rashid. Convergence Analysis of Three-Level Fourier Pseudospectral Method for Korteweg-de Vries Burgers Equation. Computers and Mathematics with Applications, 2006,52:769-778
35 M. Gulsu. A Finite Difference Approach for Solution of Burgers’Equation. Applied Mathematics and Computation,2006,175:1245-1255
36 M. Gulsu, T. Ozis. Numerical Solution of Burgers’Equation with Restrictive Taylor Approximation. Applied Mathematics and Computation,2005,171: 1192-1200
37 I. A. Hassanien, A. A. Salama, H. A. Hosham. Fourth-order Finite Difference Method for Solving Burgers’Equation. Applied Mathematics and Computation,2005,170: 781-800
38 K. K. Mohan, K. K. Sharma, A. Awasthi. A Parameter-uniform Implicit Difference Scheme for Solving Time-dependent Burgers’Equations. Applied Mathematics and Computation,2005,170:1365-1393
39左进明,周运明.一类组合Kdv-Burgers方程的数值解法.山东大学学报(理学版),2006,41(4):48-52
40张天德,鲁统超,左进明等.高阶广义Kdv方程和Kp方程的数值解法.系统科学与数学,2005,25(6):658-668
41 J. C. Eilbeck, G. R. Mcguire. Numerical Study of the Regularized Long-waveEquation, I: Numerical Methods. J. Comput. Phys.1975,19:43-57
42 J. C. Eilbeck, G. R. Mcguire. Numerical Study of the Regularized Long-wave Equation, II: Numerical Methods. J. Comput. Phys.1977,23:63-73
43 Q. Chang, G. Wang, B. Guo. Conservative Scheme for a Model of Nonlinear Dispersive Waves and its Solitary Waves Induced by Boundary Notion. J. Comput. Phys.1991,93:360-375
44 L. Zhang, Q. Chang. A New Finite Difference Method for Regularized Long-wave Equation. Chinese J. Numer. Method Comput. Appl,2001,23:58-66
45 D. Bhardwaj, R. Shankar. A Computational Method for Regularized Long Wave Equation. Comput. Math. Appl,2000,40:1397-1404
46 K. Dogan. A Numerical Simulation of Solitary-wave Solutions of the Generalized Regularized Long-wave Equation, Appl. Math.Comput,2004,149:833-841
47 A. Harten. High Resolution Schemes for Hyperbolic Conservation Laws. J Comput Phys,1983,49:357-393
48张涵信.无波动、无自由参数的耗散差分格式.空气动力学学报,1988,6(2):143-165
49 C. W. Shu, S. Osher. Efficient Implementation of Essentially Non-oscillatory Shock-capturing Schemes. J Comput Phys,1988,77:439-471
50 Y. L. Zhou. Application of Discrete Functional Analysis to the Finite Difference Method. International Academic Publishers,Beijing,1990:7-24
51张诚坚,高健,何南忠.计算方法.高等教育出版社,施普林格出版社,2003,52-53
52 J. S. Russell. Report on Waves. Proc Roy Soc,1844,2:319-320
53 D. J. Korteweg, G. D. Vires. On the Change of Form of Long Waves Advancing in a Rectangular Channel, and a New Type of Long Stationary Wave. Phil Mag,1895, 5(39): 422-443
54徐昌智,郑春龙.Kdv-Burgers方程和Schrodinger-Kdv耦合方程的显式行波解.贵州师范大学学报,2003,21(3):43-47
55 J. L. Bona, W. G. Pritchard, L. R. Scott. An Evaluation of a Model Equation for Water Waves. Phil. Trans. Roy. Soc. London,1981,302(1471):457-510
56 M. Wang. Exact Solutions for the RLW-Burgers Equation. Chinese J. Math.Appl,1995,8(1):51-55
57 D. H. Peregrine. Calculations of the Development of an Undular Bore. J. Fluid mech,1966,25:321-330
58 T. B. Benjamin, J. L. Bona, J. J. Mahony. Model Equations for Long Waves in Nonlinear Dispersive Systems. philos, Trans. Roy. Soc. London Ser. A,1972,272: 47-78
59 M. Mei. Lq -Decay Rates of Solutions for Benjamin-Bona-Mahony-Burgers Equations. J. Journal of Differential Equations,1999,158:314-340
2 L. Zhang. A Finite Difference Scheme for Generalized Regularized Long-wave Equation. Appl. Math. Comput,2005,168:962-972
3李荣华.偏微分方程数值解法.高等教育出版社,2006:13-183
4陆金甫,关治.偏微分方程数值解法.清华大学出版社,2005:45-55
5张文生.科学计算中的偏微分方程有限差分法.高等教育出版社,2006:275-282
6 V. V. Rusanov. On Difference Schemes of Third Order Accuracy for Nonlinear Hyperbolic Systems. J Comput Phys,1970,5:507-516
7 S. Z. Burstein, A. A. Mirin. Third Order Difference Methods for Hyperbolic Equations. J Comput Phys,1970,5:547-571
8 R. F. Warming, P. Kutler, H. Lomax. Second and Third-order Noncentered Difference Schemes for Nonlinear Hyperbolic Equations. AIAAJ,1973,11:189-196
9赵广慧,陶建华.浅水非线性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
10 S. K. Lele. Compact Finite Difference Schemes with Spectral-like Resolution. J Comput Phys,1992,103:16-42
11 B. Cockbum, C. W. Shu. Nonlinearly Stable Compact Schemes for Shock Caculations. SIAM J Numer Anal,1994,31(3):607-627
12 X. G. Deng, M. Hiroshi. Compact High-order Accurate Nonlinear Schemes. J Comput Phys,1997,130(1):77-91
13杨志峰,陈国谦.非定常对流扩散方程的无条件稳定高精度紧致差分格式.自然科学进展,1993,3(3):263-266
14杨志峰,周同明.含源扩散方程的一种高精度差分方法.北京师范大学学报(自然科学版),1992,28(1):124-128
15杨志峰,陈国谦.含源扩散方程的一种高精度差分方法(II).北京师范大学学报(自然科学版),1992,28(3):315-316
16杨志峰,许协庆.泊松方程的高精度优化差分方法.水动力学研究与进展,1992,(A) 7(3):263-269
17杨志峰,陈国谦,罗朝俊.一类高精度待定差分格式.见:王光谦,曲政,龚旗煌编,中国博士后论文集.北京:北京大学出版社,1991,(4):99-105
18杨志峰,陈国谦.含源汇非定常对流扩散问题紧致四阶差分格式.科学通报,1993, 38(2):113-116
19杨志峰.流体动力与传热的高级有限差分方法.见:国家自然科学青年基金研究成果报告会摘要汇编(力学).北京:科学出版社,1995:81-82
20陈国谦,杨志峰.Navier-Stokes方程的倒数型四阶紧致差分格式.见:王光谦,曲政,龚旗煌编,中国博士后论文集.北京:北京大学出版社,1991(4):132-141
21陈国谦,杨志峰,高智.对流扩散方程的四阶指数型差分格式.计算物理,1991,8(4): 359-372
22陈国谦,杨志峰.对流扩散方程的指数型摄动差分法.计算物理,1993,10(2):197-207
23 G. Q. Chen, Z. Gao, Z. F. Yang. A Pertubational h 4 Exponential Finite Difference Scheme for the Convective Diffusion Equation. J Comput Phys,1993,104(1):129-139
24 G. Q. Chen, Z. F. Yang. A Peturbational Fourth-order Upwind Finite Difference Scheme for the Convection-diffusion Equation. Journal of Hydrodynamics,1993,(B), 5(1): 82-97
25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
26 R. S. Hirsh. Higher Order Accurate Difference Solutions of Fluid Mechanics Problems by a Compact Differencing Technique. J Comput Phys,1975,19(1):90-109
27 I. Dag, D. Irk, B. Saka. A Numerical Solution of the Burger’s Equation Using Cubic B-splines. Applied Mathematics and Computation,2005,163:199-211
28 B. A. Refik, M. A. Saglam. A Mixed Finite Difference and Boundary Element Approach to One-dimensional Burgers’Equation. Applied Mathematics and Computation,2005, 160:663-673
29 E. M. Ronald. A Nonstandard Finite Difference Scheme for the Diffusionless BurgersEquation with Logistic Reaction. Mathematics and Computers in Simulation, 2003,62:117-124
30 M. A. Hela, M. S. Mehanna. A Comparison between Two Different Methods for Solving Kdv-Burgers Equation. Chaos, Solitons and Fractals,2006,28:320-326
31彭亚绵,闵涛,张世梅等.Burgers方程的MOL数值解法.西安理工大学学报,2004,20 (3):276-279
32 H. A. Amir, E. Shakour, M. Dehghan. Some Implicit Methods for the Numerical Solution of Burgers’Equation. Applied Mathematics and Computation,2007, 191:560-570
33 K. K. Mohan, A. Awasthi. A Numerical Method Based on Crank-Nicolson Scheme for Burgers’Equation. Applied Mathematics and Computation,2006,182:1430-1442
34 A. Rashid. Convergence Analysis of Three-Level Fourier Pseudospectral Method for Korteweg-de Vries Burgers Equation. Computers and Mathematics with Applications, 2006,52:769-778
35 M. Gulsu. A Finite Difference Approach for Solution of Burgers’Equation. Applied Mathematics and Computation,2006,175:1245-1255
36 M. Gulsu, T. Ozis. Numerical Solution of Burgers’Equation with Restrictive Taylor Approximation. Applied Mathematics and Computation,2005,171: 1192-1200
37 I. A. Hassanien, A. A. Salama, H. A. Hosham. Fourth-order Finite Difference Method for Solving Burgers’Equation. Applied Mathematics and Computation,2005,170: 781-800
38 K. K. Mohan, K. K. Sharma, A. Awasthi. A Parameter-uniform Implicit Difference Scheme for Solving Time-dependent Burgers’Equations. Applied Mathematics and Computation,2005,170:1365-1393
39左进明,周运明.一类组合Kdv-Burgers方程的数值解法.山东大学学报(理学版),2006,41(4):48-52
40张天德,鲁统超,左进明等.高阶广义Kdv方程和Kp方程的数值解法.系统科学与数学,2005,25(6):658-668
41 J. C. Eilbeck, G. R. Mcguire. Numerical Study of the Regularized Long-waveEquation, I: Numerical Methods. J. Comput. Phys.1975,19:43-57
42 J. C. Eilbeck, G. R. Mcguire. Numerical Study of the Regularized Long-wave Equation, II: Numerical Methods. J. Comput. Phys.1977,23:63-73
43 Q. Chang, G. Wang, B. Guo. Conservative Scheme for a Model of Nonlinear Dispersive Waves and its Solitary Waves Induced by Boundary Notion. J. Comput. Phys.1991,93:360-375
44 L. Zhang, Q. Chang. A New Finite Difference Method for Regularized Long-wave Equation. Chinese J. Numer. Method Comput. Appl,2001,23:58-66
45 D. Bhardwaj, R. Shankar. A Computational Method for Regularized Long Wave Equation. Comput. Math. Appl,2000,40:1397-1404
46 K. Dogan. A Numerical Simulation of Solitary-wave Solutions of the Generalized Regularized Long-wave Equation, Appl. Math.Comput,2004,149:833-841
47 A. Harten. High Resolution Schemes for Hyperbolic Conservation Laws. J Comput Phys,1983,49:357-393
48张涵信.无波动、无自由参数的耗散差分格式.空气动力学学报,1988,6(2):143-165
49 C. W. Shu, S. Osher. Efficient Implementation of Essentially Non-oscillatory Shock-capturing Schemes. J Comput Phys,1988,77:439-471
50 Y. L. Zhou. Application of Discrete Functional Analysis to the Finite Difference Method. International Academic Publishers,Beijing,1990:7-24
51张诚坚,高健,何南忠.计算方法.高等教育出版社,施普林格出版社,2003,52-53
52 J. S. Russell. Report on Waves. Proc Roy Soc,1844,2:319-320
53 D. J. Korteweg, G. D. Vires. On the Change of Form of Long Waves Advancing in a Rectangular Channel, and a New Type of Long Stationary Wave. Phil Mag,1895, 5(39): 422-443
54徐昌智,郑春龙.Kdv-Burgers方程和Schrodinger-Kdv耦合方程的显式行波解.贵州师范大学学报,2003,21(3):43-47
55 J. L. Bona, W. G. Pritchard, L. R. Scott. An Evaluation of a Model Equation for Water Waves. Phil. Trans. Roy. Soc. London,1981,302(1471):457-510
56 M. Wang. Exact Solutions for the RLW-Burgers Equation. Chinese J. Math.Appl,1995,8(1):51-55
57 D. H. Peregrine. Calculations of the Development of an Undular Bore. J. Fluid mech,1966,25:321-330
58 T. B. Benjamin, J. L. Bona, J. J. Mahony. Model Equations for Long Waves in Nonlinear Dispersive Systems. philos, Trans. Roy. Soc. London Ser. A,1972,272: 47-78
59 M. Mei. Lq -Decay Rates of Solutions for Benjamin-Bona-Mahony-Burgers Equations. J. Journal of Differential Equations,1999,158:314-340