Solving the systems of equations arising in the discretization of some nonlinear p.d.e.'s by implicit Runge-Kutta methods

On construit et analyse des methodes iteratives permettant une resolution efficace des systemes non lineaires issus de la discretisation en temps d'equations d'evolution non lineaires par des methodes de Range-Kutta implicites. Certains schemas consideres derivent de la methode de Newton et s'appliquent a une large classe d'equations non lineaires.

[1]  J. Butcher.  The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods , 1987 .

[2]  Ohannes A. Karakashian,et al.  Convergence of Galerkin Approximations for the Korteweg-de Vries Equation, , 1983 .

[3]  R. Alexander,et al.  The modified Newton method in the solution of stiff ordinary differential equations , 1991 .

[4]  Michel Crouzeix,et al.  On the existence of solutions to the algebraic equations in implicit runge-kutta methods , 1983 .

[5]  E. Hairer,et al.  Solving Ordinary Differential Equations I , 1987 .

[6]  F. Smith,et al.  Conservative, high-order numerical schemes for the generalized Korteweg—de Vries equation , 1995, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[7]  Georgios Akrivis,et al.  On optimal order error estimates for the nonlinear Schro¨dinger equation , 1993 .

[8]  Ohannes A. Karakashian,et al.  On some high-order accurate fully discrete Galerkin methods for the Korteweg-de Vries equation , 1985 .

[9]  G. Akrivis,et al.  On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation , 1991 .

[10]  A new class of results for the algebraic equations of implicit Runge-Kutta processes , 1991 .

[11]  Ohannes A. Karakashian,et al.  On optimal high-order in time approximations for the Korteweg-de Vries equation , 1990 .

[12]  Vidar Thomée,et al.  Convergence Estimates for Galerkin Methods for Variable Coefficient Initial Value Problems , 1974 .

[13]  O. Karakashian,et al.  On the parallel implementation of implicit Runge-Kutta methods , 1988 .

[14]  J. Bona,et al.  The initial-value problem for the Korteweg-de Vries equation , 1975, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.