Abstract
We consider implicit integration methods for the solution of stiff initial value problems for second-order differential equations of the special form y'' = f(y). In implicit methods, we are faced with the problem of solving systems of implicit relations. This paper focuses on the construction and analysis of iterative solution methods which are effective in cases where the Jacobian of the right‐hand side of the differential equation can be split into a sum of matrices with a simple structure. These iterative methods consist of the modified Newton method and an iterative linear solver to deal with the linear Newton systems. The linear solver is based on the approximate factorization of the system matrix associated with the linear Newton systems. A number of convergence results are derived for the linear solver in the case where the Jacobian matrix can be split into commuting matrices. Such problems often arise in the spatial discretization of time‐dependent partial differential equations. Furthermore, the stability matrix and the order of accuracy of the integration process are derived in the case of a finite number of iterations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.C. Butcher, The Numerical Analysis of Ordinary Differential Equations, Runge–Kutta and General Linear Methods (Wiley, New York, 1987).
C.A. Coulson, Waves, a Mathematical Account of the Common Types of Wave Motion (Oliver and Boyd, Edinburgh, 1958).
C. Eichler-Liebenow, P.J. van der Houwen and B.P. Sommeijer, Analysis of approximate factorization in iteration methods, Appl. Numer. Math. 28 (1998) 245–258.
G.H. Golub and C.F. van Loan, Matrix Computations (North Oxford Academic, 1989).
E. Hairer, Unconditionally stable methods for second-order differential equations, Numer. Math. 32 (1979) 373–379.
E. Hairer and G. Wanner, Solving Ordinary Differential Equations, Vol. II. Stiff and Differential–Algebraic Problems (Springer, Berlin, 1991).
P.J. van der Houwen, B.P. Sommeijer and J. Kok, The iterative solution of fully implicit discretizations of 3-dimensional transport models, Appl. Numer. Math. 25 (1997) 243–256.
D.W. Peaceman and H.H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math. 3 (1955) 28–41.
L.F. Shampine, Implementation of implicit formulas for the solution of ODEs, SIAM J. Sci. Statist. Comput. 1 (1980) 103–118.
P.W. Sharp, J.H. Fine and K. Burrage, Two-stage and three-stage diagonally implicit Runge–Kutta–Nyström methods of orders three and four, IMA J. Numer. Anal. 10 (1990) 489–504.
C.B. Vreugdenhil, Numerical Methods for Shallow-Water Flow (Kluwer Academic, Dordrecht, 1994).
A.G. Webster, Partial Differential Equations of Mathematical Physics (Dover, New York, 1933).
Rights and permissions
About this article
Cite this article
van der Houwen, P., Messina, E. Splitting methods for second‐order initial value problems. Numerical Algorithms 18, 233–257 (1998). https://doi.org/10.1023/A:1019173532665
Issue Date:
DOI: https://doi.org/10.1023/A:1019173532665