A high order implicit algorithm for solving instationary non-linear problems

 In this paper a high order implicit algorithm is developed for solving instationary non-linear problems. This generic numerical method combines four mathematical techniques: a time discretization, a homotopy transformation, a perturbation technique and a space discretization. The time integration is performed by classical implicit schemes (Euler implicit for problems with a first order time derivative and Newmark for second order). The time-discretization leads to non-linear equations. In this paper a new technique is proposed to solve iteratively the latter equations. The key points in this approach are, first a high order solver based on perturbation techniques, second the possibility of choosing the iteration operator, which limits the number of matrices to be triangulated. To illustrate the performance of the proposed algorithm two examples are considered: the Korteweg-de Vries equation (KdV) and the non-linear oscillations of a 2D elastic pendulum. Received 10 February 2001 / Accepted 19 December 2001