求解刚性问题的叠加Runge-Kutta方法的B-收敛性
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摘要
刚性问题是一类特殊的微分方程初值问题，具有广泛的应用背景．而Runge-Kutta方法是一种求解微分方程初值问题的常用的方法．对于刚性问题，我们一般采用隐式Runge-Kutta方法计算，这样我们可以得到高精度的数值解，但是要以很大的计算量为代价．因此，我们经常采用对角隐式Runge-Kutta方法来计算，以减少计算量．近年来，越来越多的学者对可分为两部分甚至更多的部分刚性问题产生了浓厚的兴趣．对于这类刚性问题的求解，为了减少计算量，我们采用叠加Runge-Kutta方法：对于刚性部分我们用隐式的Runge-Kutta方法；对于非刚性部分可用显式的Runge-Kutta方法．
本文就是用叠加Runge-Kutta方法求解上述刚性问题，证明了代数稳定且对角稳定的叠加Runge-Kutta方法关于K_(0,0)类初值问题的最佳B-收敛阶不低于级阶，并获得了方法的最佳B-收敛阶比级阶高一的充分条件，也证明了弱代数稳定且对角稳定(或ANS-稳定)的叠加Runge-Kutta方法关于K_(0,ω)类初值问题的最佳B-收敛阶不低于级阶，并获得了方法的最佳B-收敛阶比级阶高一的充分条件．同时也证明了分数步方法Runge-Kutta方法关于K_(0,0)类、K_(0,ω)类初值问题的B-收敛性．最后证明了(θ，(?)，(?))-代数稳定且对角稳定(或ANS-稳定)的叠加Runge-Kutta方法关于K_(σ,τ)类初值问题的最佳B-收敛阶不低于级阶，并获得了方法的最佳B-收敛阶比级阶高一的充分条件．同时也证明(θ，(?),(?))-代数稳定的叠加Runge-Kutta方法关于K_(φ,(?))类初值问题的单调性和分数步方法Runge-Kutta方法关于K_(σ,τ)类初值问题的B-收敛性．
The stiff problems is a class of special initial-value problems for differential equations and have the widespread application background. But Runge-Kutta methods are used usually to solve the initial-value problems for differential equations. We often solve the stiff problems with the implicit Runge-Kutta methods. Although we can obtain the high-accuracy numerical solution, we pay large computational cost. In order to reduce the computational cost, we often use the diagonally-implicit Runge-Kutta methods to solve the initial-value problems. In recent years, much interest has been devoted to numerical integration of nonlinear stiff problems defined by operator that may be decomposed into a sum of two or more parts. To solve this class of stiff problems, we use additive Runge-Kutta methods. It is common to combine an implicit Runge-Kutta methods for stiff parts with explicit Runge-Kutta methods for nonstiff parts.
In this article, we use the additive Runge-Kutta methods to solve the above stiff problems. We show that the order of optimal B-convergence of algebraically-stable and diagonally-stable additive Runge-Kutta methods is equal at least to the stage order for K_(0,0) class initial value problems, and provide some sufficient conditions under which the order of optimal B-convergence is one higher than stage order. We also show that the order of optimal B-convergence of algebraically-stable and diagonally-stable(ANS-stable) additive Runge-Kutta methods is equal at least to the stage order for K_(0,ω) class initial value problems, and provide some sufficient conditions under which the order of optimal B-convergence is one higher than stage order. We show the optimal B-convergence of fractional step Runge-Kutta methods for K_(0,0)、K_(0,ω) class initial value problems. At last, we show that the order of optimal B-convergence of (θ, (p|-), (q|-))-algebraically-stable and diagonally-stable(ANS-stable) additive Runge-Kutta methods is equal at least to the stage order for K_(σ,τ) class initial value problems, and provide some sufficient conditions under which the order of optimal B-convergence is one higher than stage order. We show that the monotonicity of (θ, (p|-), (q|-))-algebraically-stable and the optimal B-convergence of fractional step Runge-Kutta methods for K_(Φ,φ) class initial value problems.
本文就是用叠加Runge-Kutta方法求解上述刚性问题，证明了代数稳定且对角稳定的叠加Runge-Kutta方法关于K_(0,0)类初值问题的最佳B-收敛阶不低于级阶，并获得了方法的最佳B-收敛阶比级阶高一的充分条件，也证明了弱代数稳定且对角稳定(或ANS-稳定)的叠加Runge-Kutta方法关于K_(0,ω)类初值问题的最佳B-收敛阶不低于级阶，并获得了方法的最佳B-收敛阶比级阶高一的充分条件．同时也证明了分数步方法Runge-Kutta方法关于K_(0,0)类、K_(0,ω)类初值问题的B-收敛性．最后证明了(θ，(?)，(?))-代数稳定且对角稳定(或ANS-稳定)的叠加Runge-Kutta方法关于K_(σ,τ)类初值问题的最佳B-收敛阶不低于级阶，并获得了方法的最佳B-收敛阶比级阶高一的充分条件．同时也证明(θ，(?),(?))-代数稳定的叠加Runge-Kutta方法关于K_(φ,(?))类初值问题的单调性和分数步方法Runge-Kutta方法关于K_(σ,τ)类初值问题的B-收敛性．
The stiff problems is a class of special initial-value problems for differential equations and have the widespread application background. But Runge-Kutta methods are used usually to solve the initial-value problems for differential equations. We often solve the stiff problems with the implicit Runge-Kutta methods. Although we can obtain the high-accuracy numerical solution, we pay large computational cost. In order to reduce the computational cost, we often use the diagonally-implicit Runge-Kutta methods to solve the initial-value problems. In recent years, much interest has been devoted to numerical integration of nonlinear stiff problems defined by operator that may be decomposed into a sum of two or more parts. To solve this class of stiff problems, we use additive Runge-Kutta methods. It is common to combine an implicit Runge-Kutta methods for stiff parts with explicit Runge-Kutta methods for nonstiff parts.
In this article, we use the additive Runge-Kutta methods to solve the above stiff problems. We show that the order of optimal B-convergence of algebraically-stable and diagonally-stable additive Runge-Kutta methods is equal at least to the stage order for K_(0,0) class initial value problems, and provide some sufficient conditions under which the order of optimal B-convergence is one higher than stage order. We also show that the order of optimal B-convergence of algebraically-stable and diagonally-stable(ANS-stable) additive Runge-Kutta methods is equal at least to the stage order for K_(0,ω) class initial value problems, and provide some sufficient conditions under which the order of optimal B-convergence is one higher than stage order. We show the optimal B-convergence of fractional step Runge-Kutta methods for K_(0,0)、K_(0,ω) class initial value problems. At last, we show that the order of optimal B-convergence of (θ, (p|-), (q|-))-algebraically-stable and diagonally-stable(ANS-stable) additive Runge-Kutta methods is equal at least to the stage order for K_(σ,τ) class initial value problems, and provide some sufficient conditions under which the order of optimal B-convergence is one higher than stage order. We show that the monotonicity of (θ, (p|-), (q|-))-algebraically-stable and the optimal B-convergence of fractional step Runge-Kutta methods for K_(Φ,φ) class initial value problems.
引文
[1]肖烨.几类刚性问题数值方法的收敛性[D]．湘潭大学硕士论文.
[2] Araujo A. A note on B-stability of splitting methods[J]. Computing Visual. Sci. 2004, 26(2-3): 53-57.
[3] Burrage K, Hundsdorfer J C, Verwer J G. A study of B-convergence of RungeKuttamethods[J]. Computing, 1986, 36: 17-34.
[4] Berta García-Celayeta, Inmaculada Higueras, Teo Roldan. Contractivity/monnonicity for additive Runge-Kutta methods: Inner product norms. Applied Numerical Mathematics, 2006, 56: 862-878.
[5] Dekker K, Kraaijevanger J F B M, Spijker M N. The order of B-convergence of Gaussian Runge-Kutta method[J]. Computing, 1986, 36: 35-41.
[6] Kraaijevanger J F B M. B-convergence of the implicit midpoint rule and the trapezoidal rule[J]. BIT, 1985, 25. 652-666.
[7] Burrage K, Hundsdorfer W H. The order of B-convergence of algebraically stable Runge-Kutta methods[J]. BIT, 1987, 27 . 62-71.
[8]肖爱国．Runge-Kutta方法的B-收敛阶[J]．湘潭大学自然科学学抿,1992，14(2):16-19. (Xiao Aiguo. The order of B-convergence of Runge-Kutta methods[J]. Natural Sci. J. of Xiangtan University, 1992, 14(2): 16-19).
[9] Bujanda B J C. Stablity results for fractional step discretization of time dependentcoefficient evolutionary problems[J]. Appl. Numer. Math., 2001, 38: 69-85.
[10]李寿佛著.刚性微分方程算法理论[M]．湖南科技出版社1997．
[11]刘轩黄，彭守权译、广义逆的理论和应用[M]．华中理工大学出版社.1988．
[12]王松桂，杨振海著.广义逆矩阵及应用[M]．北京工业大学出版社.1996．
[13] Kennedy C A, Carpenter M H. Additive Runge-Kutta schemes for convectiondiffussion-rection equations[J]. Appl. Numer. Math., 2003, 44(1-2): 139-181.
[14] Levy D, Tadmor E. From semidiscrete to fully discrete: stability of RungeKuttaschemes by the energy methods[J]. SIAM Rev., 1998, 40: 40-73.
[15] Pareschi L, Russo G. Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations[M]. in: Recent Trends in Numberical Analysis, Adv. Theory Comput. Math., 2001, 3: 269-288.
[16] Shen J, Zhong X. Semi-implicit Runge-Kutta schemes for non-autonomous Differential Equations in Reactive Flow Computions[C], AIAA Paper 96-1969, AIAA, Fluid Dynamics Conference, 27th, New Orleans, LA, June 17-20.
[17] Spijker M. Stepsize restrictions for stability of one-step methods in the numericalsolution of intial value problems[J]. Math. Comp., 1985, 45: 377-392.
[18] Spijker M. A note on contractivity in the numerical solution of intial value problems[J]. BIT, 1987, 27 :424-437.
[19] Tadmor E. From Semidiscrete to Fully Discrete: Stablility of Runge-Kutta Schemes by the Energy Method II[M], Proc. Appl. Math., vol. 109, SIAM, Philadelphia, PA, 2002.
[20] Zhong X. Additive semi-implicit Runge-Kutta methods for computing high speed nonequilibrium reactive fiows[J]. J. Comput. Phys., 1996,128: 19-31.
[21] Dekker K, Verwer J G. Stablity of Runge-Kutta methods for stiff nonlinear differential equation[M]. Amsterdam: North-Holland, 1984
[22] Araujo A L, Murua A, Sanz-Serna J M. Symplectic methods based on decompositions[J]. SIAM J. Numer. Anal., 1997, 34: 1926-1947.
[23] Ascher U M, Ruuth S J, Wetton B T R. Implicit-explicit methods for timedependentpartial equations[J]. SIAM J. NUmer. Anal., 1995, 32: 797-823.
[24] Biesiadecki J J, Skeel R D. Dangers of multiple time step methods[J]. J. Comput. Phys., 1993, 109: 318-328.
[25] Bujanda B, Jorge J C. Stability results for fractional step discretization of time dependent coefficient evolutionary problems [J]. Appl. Numer. Math., 2001, 38: 69-85.
[26] Burrage K, Butcher J C. Stablity criterial for implicit Runge-Kutta methods[J]. SIAM J. Numer. Anal., 1979, 16: 46-57.
[27] Butcher J C. A stability property of implicit Runge-Kutta methods [J]. BIT, 1975, 15: 358-361 .
[28] Cooper G J, Sayfy A. Additive methods for the numerical solution of ordinary differential equations[J]. Math. Comput., 1980, 35: 1159-1172.
[29] Crouzeix M. Sul la B-stabilite des methodes de Runge-Kutta[J]. Numer. Math., 1979, 32: 75-82.
[30] Espedal M S, Karlsen K H. Numerial solution of reservoir flow methods based on large time step operator splitting algorithms. Filtration in Porous Me-??dia and Industrial Applictions. Lecture Notes in Math., No. 1734. Berlin: Springer-Verlag 2000, pp. 9-77.
[31] Hairer E, Wanner G. Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems[M]. 2nd edu., Berlin: Springer-Verlag, 1996.
[32] Lanser D, Bloom J G, Verwer J G. Time integration of the shallow water equations in spherical geometry[J]. J. Comput. Phys., 2001, 171: 1-21.
[33] McLachlan R I, Quispel G R W. Spliting methods[J]. Acta Numerica 2002, 11: 341-434.
[34] Najm H N, Wyckoff P S, Knio O M. A semi-implicit numerical scheme for reaction flow II. Stiff chemistry[J]. J. Comput. Phys., 1998,134: 381-402.
[35] Verwer J G, Hundsdorfer W, Bloom J G. Numerical time integration for air pollution models[J]. Surveys Math. Indust., 2001, 10: 107-174.
[36] Karniadakis G E, Israeli M and Orszag S A. High-order splitting methods for the incompressible Navier-stokes equations[J]. J. Comput. Phys., 1991, 97: 414-443.
[37] Kim J, Moin P. Application of a fractional-step method to incompressible Navier-stokes equations[J]. J. Comput. Phys., 1985, 59: 308-323.
[38] Ruuth S. Implicit-explicit methods for rection-diffusion problems in pattern formation[J]. J. Math. Biol, 1995, 34(2): 148-176.
[39] Skeel R D, Zhang G, Schlick T. A family of symplectic integrators: stablity, accuracy, and molecular dynamics applications [J]. SIAM J. Sci. Comput., 1997, 18(1): 203-222.
[40] Spiteri R, Ascher U, Pai D. Numerical solution of differential systems with algebraic inequalities arising in robot programming[C], in: Proceedings of IEEE Conference on Robotics and Automation (1995).
[41] Turek S. A comparative study of time stepping techniques for the incompressibleNavier-stokers equations: from fully implicit non-linear schemes to semiexplicitprojection methods[J]. Internat. J. Numer. Methods Fluids, 1996, 22: 987-1011.
[42] Varah J M. Stability restrictions on second order, three level finite difference schemes for parabolic equations [J]. SIAM J. Numer. Anal., 1980, 17(2): 300-309.
[43] Verwer J G, Blom J G, Hundsdorfer W. An implicit-explicit approach for atmospheric transport chemistry problems[J]. Appl. Numer. Math., 1996, 20: 191-209.
[44]肖爱国．非线性刚性微分方程算法理论的发展[J]．自然科学进屁1999，12(9)： 1065-1072．
[45]肖爱国．Runge-Kutta方法的((?)，p，q)代数稳定性[J]．计算数学,1993，4：440- 448．
[46] Lambert J D. Computational methods in ordinary differential equations[M]. Chichester: John Wiley k Sons Lod, 1973.
[47] Shampine L F, Gear C W. A user's view of solving stiff ordinary differtial equations[J]. SIAM Review, 1979, 21: 1-17.
[48] Dahlqust G. Error analysis for a class of methods for stiff nonlinear intial value problems . Numer Anal Dundee, Lecture Notes in Math 506. Berlin: Springer-Verlag, 1975. 60-74.
[49] Cooper G J, Sayfy A. Additive methods for the numerical solution of ordinary differential equations[J]. Math. Comput., 1983, 40: 207-218.
[50] Ascher Uri M, Ruuth Steven J, Spiteri Raymond J. Implicit-explicit RungeKuttamethods for time-dependent partal differential equations [J]. Appl. Numer.Math., 1997, 25: 151-167.
[51] Liu Hongyu, Zou Jun. Some new additive Runge-Kutta methods and their applications[J]. J. Comput. Appl. Math., 2006, 190: 74-98.
[52] Lorenzo Pareschi, Giovanni Russo. Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation [J]. Journal of Scientific Computing, 2005, 25: 129-155.
[2] Araujo A. A note on B-stability of splitting methods[J]. Computing Visual. Sci. 2004, 26(2-3): 53-57.
[3] Burrage K, Hundsdorfer J C, Verwer J G. A study of B-convergence of RungeKuttamethods[J]. Computing, 1986, 36: 17-34.
[4] Berta García-Celayeta, Inmaculada Higueras, Teo Roldan. Contractivity/monnonicity for additive Runge-Kutta methods: Inner product norms. Applied Numerical Mathematics, 2006, 56: 862-878.
[5] Dekker K, Kraaijevanger J F B M, Spijker M N. The order of B-convergence of Gaussian Runge-Kutta method[J]. Computing, 1986, 36: 35-41.
[6] Kraaijevanger J F B M. B-convergence of the implicit midpoint rule and the trapezoidal rule[J]. BIT, 1985, 25. 652-666.
[7] Burrage K, Hundsdorfer W H. The order of B-convergence of algebraically stable Runge-Kutta methods[J]. BIT, 1987, 27 . 62-71.
[8]肖爱国．Runge-Kutta方法的B-收敛阶[J]．湘潭大学自然科学学抿,1992，14(2):16-19. (Xiao Aiguo. The order of B-convergence of Runge-Kutta methods[J]. Natural Sci. J. of Xiangtan University, 1992, 14(2): 16-19).
[9] Bujanda B J C. Stablity results for fractional step discretization of time dependentcoefficient evolutionary problems[J]. Appl. Numer. Math., 2001, 38: 69-85.
[10]李寿佛著.刚性微分方程算法理论[M]．湖南科技出版社1997．
[11]刘轩黄，彭守权译、广义逆的理论和应用[M]．华中理工大学出版社.1988．
[12]王松桂，杨振海著.广义逆矩阵及应用[M]．北京工业大学出版社.1996．
[13] Kennedy C A, Carpenter M H. Additive Runge-Kutta schemes for convectiondiffussion-rection equations[J]. Appl. Numer. Math., 2003, 44(1-2): 139-181.
[14] Levy D, Tadmor E. From semidiscrete to fully discrete: stability of RungeKuttaschemes by the energy methods[J]. SIAM Rev., 1998, 40: 40-73.
[15] Pareschi L, Russo G. Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations[M]. in: Recent Trends in Numberical Analysis, Adv. Theory Comput. Math., 2001, 3: 269-288.
[16] Shen J, Zhong X. Semi-implicit Runge-Kutta schemes for non-autonomous Differential Equations in Reactive Flow Computions[C], AIAA Paper 96-1969, AIAA, Fluid Dynamics Conference, 27th, New Orleans, LA, June 17-20.
[17] Spijker M. Stepsize restrictions for stability of one-step methods in the numericalsolution of intial value problems[J]. Math. Comp., 1985, 45: 377-392.
[18] Spijker M. A note on contractivity in the numerical solution of intial value problems[J]. BIT, 1987, 27 :424-437.
[19] Tadmor E. From Semidiscrete to Fully Discrete: Stablility of Runge-Kutta Schemes by the Energy Method II[M], Proc. Appl. Math., vol. 109, SIAM, Philadelphia, PA, 2002.
[20] Zhong X. Additive semi-implicit Runge-Kutta methods for computing high speed nonequilibrium reactive fiows[J]. J. Comput. Phys., 1996,128: 19-31.
[21] Dekker K, Verwer J G. Stablity of Runge-Kutta methods for stiff nonlinear differential equation[M]. Amsterdam: North-Holland, 1984
[22] Araujo A L, Murua A, Sanz-Serna J M. Symplectic methods based on decompositions[J]. SIAM J. Numer. Anal., 1997, 34: 1926-1947.
[23] Ascher U M, Ruuth S J, Wetton B T R. Implicit-explicit methods for timedependentpartial equations[J]. SIAM J. NUmer. Anal., 1995, 32: 797-823.
[24] Biesiadecki J J, Skeel R D. Dangers of multiple time step methods[J]. J. Comput. Phys., 1993, 109: 318-328.
[25] Bujanda B, Jorge J C. Stability results for fractional step discretization of time dependent coefficient evolutionary problems [J]. Appl. Numer. Math., 2001, 38: 69-85.
[26] Burrage K, Butcher J C. Stablity criterial for implicit Runge-Kutta methods[J]. SIAM J. Numer. Anal., 1979, 16: 46-57.
[27] Butcher J C. A stability property of implicit Runge-Kutta methods [J]. BIT, 1975, 15: 358-361 .
[28] Cooper G J, Sayfy A. Additive methods for the numerical solution of ordinary differential equations[J]. Math. Comput., 1980, 35: 1159-1172.
[29] Crouzeix M. Sul la B-stabilite des methodes de Runge-Kutta[J]. Numer. Math., 1979, 32: 75-82.
[30] Espedal M S, Karlsen K H. Numerial solution of reservoir flow methods based on large time step operator splitting algorithms. Filtration in Porous Me-??dia and Industrial Applictions. Lecture Notes in Math., No. 1734. Berlin: Springer-Verlag 2000, pp. 9-77.
[31] Hairer E, Wanner G. Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems[M]. 2nd edu., Berlin: Springer-Verlag, 1996.
[32] Lanser D, Bloom J G, Verwer J G. Time integration of the shallow water equations in spherical geometry[J]. J. Comput. Phys., 2001, 171: 1-21.
[33] McLachlan R I, Quispel G R W. Spliting methods[J]. Acta Numerica 2002, 11: 341-434.
[34] Najm H N, Wyckoff P S, Knio O M. A semi-implicit numerical scheme for reaction flow II. Stiff chemistry[J]. J. Comput. Phys., 1998,134: 381-402.
[35] Verwer J G, Hundsdorfer W, Bloom J G. Numerical time integration for air pollution models[J]. Surveys Math. Indust., 2001, 10: 107-174.
[36] Karniadakis G E, Israeli M and Orszag S A. High-order splitting methods for the incompressible Navier-stokes equations[J]. J. Comput. Phys., 1991, 97: 414-443.
[37] Kim J, Moin P. Application of a fractional-step method to incompressible Navier-stokes equations[J]. J. Comput. Phys., 1985, 59: 308-323.
[38] Ruuth S. Implicit-explicit methods for rection-diffusion problems in pattern formation[J]. J. Math. Biol, 1995, 34(2): 148-176.
[39] Skeel R D, Zhang G, Schlick T. A family of symplectic integrators: stablity, accuracy, and molecular dynamics applications [J]. SIAM J. Sci. Comput., 1997, 18(1): 203-222.
[40] Spiteri R, Ascher U, Pai D. Numerical solution of differential systems with algebraic inequalities arising in robot programming[C], in: Proceedings of IEEE Conference on Robotics and Automation (1995).
[41] Turek S. A comparative study of time stepping techniques for the incompressibleNavier-stokers equations: from fully implicit non-linear schemes to semiexplicitprojection methods[J]. Internat. J. Numer. Methods Fluids, 1996, 22: 987-1011.
[42] Varah J M. Stability restrictions on second order, three level finite difference schemes for parabolic equations [J]. SIAM J. Numer. Anal., 1980, 17(2): 300-309.
[43] Verwer J G, Blom J G, Hundsdorfer W. An implicit-explicit approach for atmospheric transport chemistry problems[J]. Appl. Numer. Math., 1996, 20: 191-209.
[44]肖爱国．非线性刚性微分方程算法理论的发展[J]．自然科学进屁1999，12(9)： 1065-1072．
[45]肖爱国．Runge-Kutta方法的((?)，p，q)代数稳定性[J]．计算数学,1993，4：440- 448．
[46] Lambert J D. Computational methods in ordinary differential equations[M]. Chichester: John Wiley k Sons Lod, 1973.
[47] Shampine L F, Gear C W. A user's view of solving stiff ordinary differtial equations[J]. SIAM Review, 1979, 21: 1-17.
[48] Dahlqust G. Error analysis for a class of methods for stiff nonlinear intial value problems . Numer Anal Dundee, Lecture Notes in Math 506. Berlin: Springer-Verlag, 1975. 60-74.
[49] Cooper G J, Sayfy A. Additive methods for the numerical solution of ordinary differential equations[J]. Math. Comput., 1983, 40: 207-218.
[50] Ascher Uri M, Ruuth Steven J, Spiteri Raymond J. Implicit-explicit RungeKuttamethods for time-dependent partal differential equations [J]. Appl. Numer.Math., 1997, 25: 151-167.
[51] Liu Hongyu, Zou Jun. Some new additive Runge-Kutta methods and their applications[J]. J. Comput. Appl. Math., 2006, 190: 74-98.
[52] Lorenzo Pareschi, Giovanni Russo. Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation [J]. Journal of Scientific Computing, 2005, 25: 129-155.