求解刚性振荡问题的对角隐式Runge-Kutta方法
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摘要
刚性振荡问题常出现在现代科学技术的许多领域,其数值方法的研究具有广泛的应用前景.由于刚性振荡问题具有刚性和振荡性双重特性,其高效数值求解具有一定的挑战性.多年来,许多学者一直在关注并努力获得数值求解的高效算法。
本文主要是在前人工作的基础上对求解刚性振荡问题的对角隐式Runge-Kutta方法进行研究,通过提高方法的代数阶、稳定性条件,以及对相误差和耗散误差的控制来实现方法的有效性.全文由五章组成.
第一章阐述了研究背景和本文的主要工作。
第二章介绍刚性振荡问题及其数值方法,包括2级3阶对角隐式Runge-Kutta方法、第一级为显式的3级3阶对角隐式Runge-Kutta方法、3级4阶对称对角隐式Runge-Kutta方法和第一级为显式的4级4阶对称对角隐式Runge-Kutta方法.
第三章给出了方法A-稳定时方法系数应该满足的范围.
第四章对所构造的满足阶条件和稳定性要求的方法进行相误差和耗散误差分析.
第五章通过求解实际的刚性振荡问题,来验证所构造的方法的有效性.
Stiff oscillatory problems axe involved in various fields of modern science and technology. The research on their numerical methods has wide prospect in applications. Due to its two-sided characteristics, namely, stiffness and oscillation, it is rather difficult and challenging to obtain highly-efficient numerical methods, which are attracting many scholars' attention for a long time.
This paper mainly consider diagonally implicit Runge-Kutta methods for solving stiff problems with oscillating solutions based on other scholars' research.We make our methods more effective by improving the algebraic order, stability conditions of the methods, and controlling the phase error and amplification error. This paper contains five parts.
In chapter 1, we introduce the background of research and main work of this paper.
In chapter 2, we introduce the considered problem and the RK methods for solving it, such as two-stage diagonally-implicit Runge-Kutta methods of order three, three-stage diagonally-implicit Runge-Kutta methods with an explicit first stage of order three, three-stage symmetric diagonally-implicit Runge-Kutta methods of order four and four-stage symmetric diagonally-implicit Runge-Kutta methods with an explicit first stage of order four.
In chapter 3, we analyze the scope which the coefficients of the methods should satisfy when the methods are A-stable.
In chapter 4, we study the phase error and amplification error of the methods to make the order of them as high as possible.
In chapter 5, we take our methods to numerical experiments. It is shown that the constructed methods are efficient.
本文主要是在前人工作的基础上对求解刚性振荡问题的对角隐式Runge-Kutta方法进行研究,通过提高方法的代数阶、稳定性条件,以及对相误差和耗散误差的控制来实现方法的有效性.全文由五章组成.
第一章阐述了研究背景和本文的主要工作。
第二章介绍刚性振荡问题及其数值方法,包括2级3阶对角隐式Runge-Kutta方法、第一级为显式的3级3阶对角隐式Runge-Kutta方法、3级4阶对称对角隐式Runge-Kutta方法和第一级为显式的4级4阶对称对角隐式Runge-Kutta方法.
第三章给出了方法A-稳定时方法系数应该满足的范围.
第四章对所构造的满足阶条件和稳定性要求的方法进行相误差和耗散误差分析.
第五章通过求解实际的刚性振荡问题,来验证所构造的方法的有效性.
Stiff oscillatory problems axe involved in various fields of modern science and technology. The research on their numerical methods has wide prospect in applications. Due to its two-sided characteristics, namely, stiffness and oscillation, it is rather difficult and challenging to obtain highly-efficient numerical methods, which are attracting many scholars' attention for a long time.
This paper mainly consider diagonally implicit Runge-Kutta methods for solving stiff problems with oscillating solutions based on other scholars' research.We make our methods more effective by improving the algebraic order, stability conditions of the methods, and controlling the phase error and amplification error. This paper contains five parts.
In chapter 1, we introduce the background of research and main work of this paper.
In chapter 2, we introduce the considered problem and the RK methods for solving it, such as two-stage diagonally-implicit Runge-Kutta methods of order three, three-stage diagonally-implicit Runge-Kutta methods with an explicit first stage of order three, three-stage symmetric diagonally-implicit Runge-Kutta methods of order four and four-stage symmetric diagonally-implicit Runge-Kutta methods with an explicit first stage of order four.
In chapter 3, we analyze the scope which the coefficients of the methods should satisfy when the methods are A-stable.
In chapter 4, we study the phase error and amplification error of the methods to make the order of them as high as possible.
In chapter 5, we take our methods to numerical experiments. It is shown that the constructed methods are efficient.
引文
[1]L.Brusa,L.Nigro.A one-step method for direct integration of structural dynamic equations[J].Internat.J.Numer.Methods Eng,15(1980) 685-699.
[2]M.Calvo,J.M.Franco,J.I.Montijano,L.Randez.Explicit Runge-Kutta methods for initial value problems with oscillating solutions[J].J.Comput.Appl.Math,76(1996) 195-212.
[3]M.M.Chawla,P.S.Rao.An explicit sixth-order method with phase-lag of order eight for y"=f(t,y)[J].J.Comput.Appl.Math,17(1987) 365-368.
[4]J.M.Franco.An explicit hybrid method of Numerov type for second-order periodic initial-value problems[J].J.Comput.Appl.Math,59(1995) 79-90.
[5]I.Gladwell,R.M.Thomas.Damping and phase analysis of some methods for solving second order ordinary differential equations[J].Internat.J.Numer.Methods.Eng,19(1983) 495-503.
[6]M.Meneguette.Chawla-Numerov method revisited.[J]J.Comput.Appl.Math,36(1991) 247-250.
[7]A.B.Sideris,T.E.Simons.A low-order embedded Runge-Kutta methods for periodic initial-value problems[J].J.Comput.Appl.Math,44(1992) 235-244.
[8]T.E.Simons.Explicit two-step methods with minimal phase-lag for the numerical integration of special second-order initial-value problems and their application to the one-dimensional Schr(o|¨)dinger equation[J].J.Comput.Appl.Math,39(1992) 89-94.
[9]R.M.Thomas.Phase properties of high-order,almost P-stable formulae[J].BIT 24(1984) 225-238.
[10]P.J.Vander Houwen,B.P.Sommeijer.Explicit Runge-Kutta(-Nystr(o|¨)m) methods with reduced phase errors for computing oscillating solutions[J].SIAM J.Numer.Anal,24(1987) 595-617.
[11]P.J.Vander Houwen,B.P.Sommeijer.Phase-lag analysis of implicit Runge-Kutta methods[J].SIAM J.Numer.Anal,26(1989) 214-229.
[12]P.J.Vander Houwen,B.P.Sommeijer.Diagonally implicit Runge-Kutta-Nystr(o|¨)m methods for oscillatory problems[J].SIAM J.Numer.Anal,26(1989) 414-429.
[13]J.M.Franco,I.Gomez,L.Randez.SDIRK methods for stiff ODEs with oscillating solutions[J].J.Comput.Appl.Math,81(1997) 197-209.
[14]J.M.Franco,I.Gomez.Fourth-order symmetric DIRK methods for periodic stiff problems[JI.Numerical Algorithms,32(2003) 317-336.
[15]A.KVAERNφ,Singly diagonally implicit Runge-Kutta methods with an explicit first stage[J].BIT Numerical Mathematics,44(2004) 489-502.
[16]张德富.A-稳定的Runge-Kutta公式及其指数拟合[D].湘潭大学硕士论文,1999.
[17]李聪颖.对角隐式龙格—库塔法及其指数拟合[D].湘潭大学硕士论文,2007.
[18]李寿佛.刚性微分方程算法理论[M].湖南科学技术出版社,1997.
[19]袁兆鼎,费景高,刘德贵,刚性常微分方程初值问题的数值方法[M].科学出版社,1987.
[20]Roger Alexander.Design and implementation of DIRK integrators for stiff systems [J].Applied Numerical Mathematics,46(2003) 1-17.
[21]P.J.Vander Houwen,B.P.Sommeijer.Phase-lag analysis of implicit Runge-Kutta methods[J].SIAM J.Numer.Anal,26(1989) 214-229.
[22]J.R.Cash.A comparison of some codes for the stiff oscillatory problem[J].Computers Math.Applic,36(1998) 51-57.
[23]L.Abahamsson,H.O.Kreiss.Explicit methods for mildly stiff oscillatory systems[J].BIT 31(1991) 608-619.
[24]I.Alonso-Mallo,B.Cano,M.J.Moreta.Stable Runge-Kutta-Nystr(o|¨)m methods for dissipative stiff problems[J].Numer Algor,42(2006) 193-203.
[25]J.C.Butcher,M.T.Diamantakis.DESIRE:diagonally extend singly implicit Runge-Kutta effective order methods[J].Numerical Algorithm 17(1998) 121-145.
[26]G.Papageorgiou,I.Th.Famelis,Ch.Tsitouras.A P-stable singly diagonally implicit Runge-Kutta-Nystr(o|¨)m method[J].Numerical Algorithms,17(1998) 345-353.
[27] M.H. Carpenter, A. Kennedy, Hester Bijl, S.A. Viken, Veer N.Vatsa. Fourth-order Runge-Kutta schemes for fluid mechanics applications[J]. Journal of Scientific Computing, 25 (2005) 10.1007/s10915-004-4637-3.
[28] Amelia Garcia, Pablo Martin, Ana B. Gonzalez. New methods for oscillatory problems based on classical codes[J]. Applied Numerical Mathematics, 42 (2002) 141-157.
[29] L.F. Shampine~1. Error estimation and control for ODEs[J]. Journal of Scientific Computing,25 (2005) 10.1007/s10915-004-4629-3.
[30] D. Fr(?)nken, K. Ochs. Passive Runge-Kutta methods-properties, Parametric representation, and order conditions[J]. BIT Numerical Mathematics, 43 (2003) 339-361.
[31] Roger Alexander. Diagonally implicit Runge-Kutta mathods for stiff ordinary differential equation[J]. SIAM J. Numer. Anal, 14 (1997) 1006-1021.
[32] E. Hairer, G. Wanner. Solving ordinary differential equations II, Stiff and Differential-Algebraic Problems[M], Springer, Berlin, 1991.
[33] Hans Van de Vyver. Stability and phase-lag analysis of explicit Runge-Kutta methods with variable coefficients for oscillatory problems[J]. Computer Physics Communications, 173 (2005) 115-130.
[34] J.C. Butcher. Implicit Runge-Kutta processes[J]. Math. Comput, 18 (1964) 50-64.
[35] J.C. Butcher. The numerical analysis of ordinary differential equations[J]. John Wiley and Sons Ltd, 1987.
[2]M.Calvo,J.M.Franco,J.I.Montijano,L.Randez.Explicit Runge-Kutta methods for initial value problems with oscillating solutions[J].J.Comput.Appl.Math,76(1996) 195-212.
[3]M.M.Chawla,P.S.Rao.An explicit sixth-order method with phase-lag of order eight for y"=f(t,y)[J].J.Comput.Appl.Math,17(1987) 365-368.
[4]J.M.Franco.An explicit hybrid method of Numerov type for second-order periodic initial-value problems[J].J.Comput.Appl.Math,59(1995) 79-90.
[5]I.Gladwell,R.M.Thomas.Damping and phase analysis of some methods for solving second order ordinary differential equations[J].Internat.J.Numer.Methods.Eng,19(1983) 495-503.
[6]M.Meneguette.Chawla-Numerov method revisited.[J]J.Comput.Appl.Math,36(1991) 247-250.
[7]A.B.Sideris,T.E.Simons.A low-order embedded Runge-Kutta methods for periodic initial-value problems[J].J.Comput.Appl.Math,44(1992) 235-244.
[8]T.E.Simons.Explicit two-step methods with minimal phase-lag for the numerical integration of special second-order initial-value problems and their application to the one-dimensional Schr(o|¨)dinger equation[J].J.Comput.Appl.Math,39(1992) 89-94.
[9]R.M.Thomas.Phase properties of high-order,almost P-stable formulae[J].BIT 24(1984) 225-238.
[10]P.J.Vander Houwen,B.P.Sommeijer.Explicit Runge-Kutta(-Nystr(o|¨)m) methods with reduced phase errors for computing oscillating solutions[J].SIAM J.Numer.Anal,24(1987) 595-617.
[11]P.J.Vander Houwen,B.P.Sommeijer.Phase-lag analysis of implicit Runge-Kutta methods[J].SIAM J.Numer.Anal,26(1989) 214-229.
[12]P.J.Vander Houwen,B.P.Sommeijer.Diagonally implicit Runge-Kutta-Nystr(o|¨)m methods for oscillatory problems[J].SIAM J.Numer.Anal,26(1989) 414-429.
[13]J.M.Franco,I.Gomez,L.Randez.SDIRK methods for stiff ODEs with oscillating solutions[J].J.Comput.Appl.Math,81(1997) 197-209.
[14]J.M.Franco,I.Gomez.Fourth-order symmetric DIRK methods for periodic stiff problems[JI.Numerical Algorithms,32(2003) 317-336.
[15]A.KVAERNφ,Singly diagonally implicit Runge-Kutta methods with an explicit first stage[J].BIT Numerical Mathematics,44(2004) 489-502.
[16]张德富.A-稳定的Runge-Kutta公式及其指数拟合[D].湘潭大学硕士论文,1999.
[17]李聪颖.对角隐式龙格—库塔法及其指数拟合[D].湘潭大学硕士论文,2007.
[18]李寿佛.刚性微分方程算法理论[M].湖南科学技术出版社,1997.
[19]袁兆鼎,费景高,刘德贵,刚性常微分方程初值问题的数值方法[M].科学出版社,1987.
[20]Roger Alexander.Design and implementation of DIRK integrators for stiff systems [J].Applied Numerical Mathematics,46(2003) 1-17.
[21]P.J.Vander Houwen,B.P.Sommeijer.Phase-lag analysis of implicit Runge-Kutta methods[J].SIAM J.Numer.Anal,26(1989) 214-229.
[22]J.R.Cash.A comparison of some codes for the stiff oscillatory problem[J].Computers Math.Applic,36(1998) 51-57.
[23]L.Abahamsson,H.O.Kreiss.Explicit methods for mildly stiff oscillatory systems[J].BIT 31(1991) 608-619.
[24]I.Alonso-Mallo,B.Cano,M.J.Moreta.Stable Runge-Kutta-Nystr(o|¨)m methods for dissipative stiff problems[J].Numer Algor,42(2006) 193-203.
[25]J.C.Butcher,M.T.Diamantakis.DESIRE:diagonally extend singly implicit Runge-Kutta effective order methods[J].Numerical Algorithm 17(1998) 121-145.
[26]G.Papageorgiou,I.Th.Famelis,Ch.Tsitouras.A P-stable singly diagonally implicit Runge-Kutta-Nystr(o|¨)m method[J].Numerical Algorithms,17(1998) 345-353.
[27] M.H. Carpenter, A. Kennedy, Hester Bijl, S.A. Viken, Veer N.Vatsa. Fourth-order Runge-Kutta schemes for fluid mechanics applications[J]. Journal of Scientific Computing, 25 (2005) 10.1007/s10915-004-4637-3.
[28] Amelia Garcia, Pablo Martin, Ana B. Gonzalez. New methods for oscillatory problems based on classical codes[J]. Applied Numerical Mathematics, 42 (2002) 141-157.
[29] L.F. Shampine~1. Error estimation and control for ODEs[J]. Journal of Scientific Computing,25 (2005) 10.1007/s10915-004-4629-3.
[30] D. Fr(?)nken, K. Ochs. Passive Runge-Kutta methods-properties, Parametric representation, and order conditions[J]. BIT Numerical Mathematics, 43 (2003) 339-361.
[31] Roger Alexander. Diagonally implicit Runge-Kutta mathods for stiff ordinary differential equation[J]. SIAM J. Numer. Anal, 14 (1997) 1006-1021.
[32] E. Hairer, G. Wanner. Solving ordinary differential equations II, Stiff and Differential-Algebraic Problems[M], Springer, Berlin, 1991.
[33] Hans Van de Vyver. Stability and phase-lag analysis of explicit Runge-Kutta methods with variable coefficients for oscillatory problems[J]. Computer Physics Communications, 173 (2005) 115-130.
[34] J.C. Butcher. Implicit Runge-Kutta processes[J]. Math. Comput, 18 (1964) 50-64.
[35] J.C. Butcher. The numerical analysis of ordinary differential equations[J]. John Wiley and Sons Ltd, 1987.