边界值方法解初值问题
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摘要
解微分方程数值解的线性k-步方法使一个一阶连续问题转化为一个k-阶离散问题，在求解k-阶离散问题时，需预先求出k个初始值，且要求满足稳定性条件。在实际计算中，k步法常常会产生假解。若把初值问题转化为一个与之等价的边值问题，就可以比较容易的控制这种假解。基于这个想法，就得到了一个新的多步方法，我们叫做边界值方法(Boundary Value Methods)，当然，边界值方法是自由边界问题。边界值方法的块方法不仅具有与R-K方法类似的格式，而且保持了线性多步法的优点，并且在区间剖分上也更加灵活。
本篇文章详细介绍了边界值方法，给出了在几个常见微分方程数值解法基础上产生的边界值方法，并且对其稳定性及收敛性作了理论分析。最后给出了几个数值例子，从初值方法与边界值方法的比较中看出边界值方法的优点。
The approximation of the solutions of ODEs by means of k-step methods transforms a first-order continuous problem in a k~(th) -order discrete one. Such transformation has the undesired effect of introducing spurious, or parasitic, solutions to be kept under control. It is such control which is responsible of the main drawbacks of the classical LMF with respect to Runge-Kutta methods. However, the control of the parasitic solutions is much easier if the problem is transformed into an almost equivalent boundary value problem. Starting from such an idea, a new class of multistep methods, called Boundary Value Methods (BVMs), has been proposed. Of course, they are free of barriers. Moreover, a block of such methods presents some similarity with Runge-Kutta schemes, although still maintaining the advantages of being linear methods, especially, the stepsize variation becomes very simple.
In this paper, we in detail introduced the boundary value method, has produced boundary value methods ETRs and GBDF which produce in classical Initial value methods, and has analyzed its stability and convergence theories. Finally we produced several numerical examples, by comparing the boundary value method and the initial value method, see the advantage of BVMs.
本篇文章详细介绍了边界值方法，给出了在几个常见微分方程数值解法基础上产生的边界值方法，并且对其稳定性及收敛性作了理论分析。最后给出了几个数值例子，从初值方法与边界值方法的比较中看出边界值方法的优点。
The approximation of the solutions of ODEs by means of k-step methods transforms a first-order continuous problem in a k~(th) -order discrete one. Such transformation has the undesired effect of introducing spurious, or parasitic, solutions to be kept under control. It is such control which is responsible of the main drawbacks of the classical LMF with respect to Runge-Kutta methods. However, the control of the parasitic solutions is much easier if the problem is transformed into an almost equivalent boundary value problem. Starting from such an idea, a new class of multistep methods, called Boundary Value Methods (BVMs), has been proposed. Of course, they are free of barriers. Moreover, a block of such methods presents some similarity with Runge-Kutta schemes, although still maintaining the advantages of being linear methods, especially, the stepsize variation becomes very simple.
In this paper, we in detail introduced the boundary value method, has produced boundary value methods ETRs and GBDF which produce in classical Initial value methods, and has analyzed its stability and convergence theories. Finally we produced several numerical examples, by comparing the boundary value method and the initial value method, see the advantage of BVMs.
引文
[1] L. Brugnano and D. Trigiante, Boundary Value Methods:The Third Way Between Linear Multistep and Runge-Kutta Methods, Computers Math. Applic. Vol. 36, No. 10-12, pp. 269-284, 1998.
[2] L. Brugnano and D. Trigiante, Convergence and stability of boundary value methods for ordinary differential equations, JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS. 66(1996) 97-109.
[3] P. Amodio and L. Brugnano, The Conditioning of Toeplitz Band Matrices, Math. Comput. Modelling Vol. 23, No. 10, pp. 29-42, 1996.
[4] J. D. Lambert, Numerical Methods for ordinary differential equations(Wiley, New York, 1991).
[5] F. IAVERNARO AND F. MAZZIA, BLOCK-BOUNDARY VALUE METHODS FOR THE SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS, SIAM J. SCI COMPUT. VOL. 21, NO. 1, PP. 323-339.
[6] F. Iavernaro, F. Mazzia, Solving ordinary differential equations by generalized Adams methods: properties and implementation techniques, Applied Numerical Mathematics 28 (1998) 107-126.
[7] Luigi Brugnano, Donato Trigiante, A new mesh selection strategy for ODEs, Applied Numerical Mathematics 24 (1997) 1-21.
[8] L. Brugnano and D. Trigiante, Solving Differential Problems by Multistep Initial and Boundary Value Methods, Gordan and Breach, Amsterdam, (1998).
[9] L. Brugnano and D. Trigiante, High order multistep methods for boundary value problem, Appl. Numer. Math. 18, 79-94 (1995).
[10] P. Amodio and F. Mazzia, A boundary value approach to the numerical solution of ODEs by multistep methods, Jour. Of Difference Eq. and Appl. 1, 353-367 (1995).
[11] P. Amodio and F. Mazzia, Boundary value methods based on Adams-type methods, Appl. Numer. Math. 18 (1-3) (1995) 23-35.
[12] A. O. H. Axelsson and J. G. Verwer, Boundary value techniques for initial value problems in ordinary differential equations, Math. Comp. 45 (1985) 153-171.
[13] P. Marzulli and D. Trigiante, On some numerical methods for solving ODE, J. Difference Eq. Appl. 1 (1995) 45-55.
[14] 余德浩，汤华中，微分方程数值解法，科学出版社，2004年6月．
[2] L. Brugnano and D. Trigiante, Convergence and stability of boundary value methods for ordinary differential equations, JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS. 66(1996) 97-109.
[3] P. Amodio and L. Brugnano, The Conditioning of Toeplitz Band Matrices, Math. Comput. Modelling Vol. 23, No. 10, pp. 29-42, 1996.
[4] J. D. Lambert, Numerical Methods for ordinary differential equations(Wiley, New York, 1991).
[5] F. IAVERNARO AND F. MAZZIA, BLOCK-BOUNDARY VALUE METHODS FOR THE SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS, SIAM J. SCI COMPUT. VOL. 21, NO. 1, PP. 323-339.
[6] F. Iavernaro, F. Mazzia, Solving ordinary differential equations by generalized Adams methods: properties and implementation techniques, Applied Numerical Mathematics 28 (1998) 107-126.
[7] Luigi Brugnano, Donato Trigiante, A new mesh selection strategy for ODEs, Applied Numerical Mathematics 24 (1997) 1-21.
[8] L. Brugnano and D. Trigiante, Solving Differential Problems by Multistep Initial and Boundary Value Methods, Gordan and Breach, Amsterdam, (1998).
[9] L. Brugnano and D. Trigiante, High order multistep methods for boundary value problem, Appl. Numer. Math. 18, 79-94 (1995).
[10] P. Amodio and F. Mazzia, A boundary value approach to the numerical solution of ODEs by multistep methods, Jour. Of Difference Eq. and Appl. 1, 353-367 (1995).
[11] P. Amodio and F. Mazzia, Boundary value methods based on Adams-type methods, Appl. Numer. Math. 18 (1-3) (1995) 23-35.
[12] A. O. H. Axelsson and J. G. Verwer, Boundary value techniques for initial value problems in ordinary differential equations, Math. Comp. 45 (1985) 153-171.
[13] P. Marzulli and D. Trigiante, On some numerical methods for solving ODE, J. Difference Eq. Appl. 1 (1995) 45-55.
[14] 余德浩，汤华中，微分方程数值解法，科学出版社，2004年6月．