Acceleration of implicit schemes for large systems of delay differential equations

Abstract

International audienceThe objective is to accelerate numerical implicit schemes for solving large linear or nonlinear delay differential equations. These schemes require solving large linear or nonlinear systems at each integration step, making effective initial guesses critical for rapid convergence. For nonlinear problems, an inexact Newton method is used, whose efficiency depends heavily on the quality of these initial guesses. To generate them, line search or trust-region algorithms are employed -each involving the solution of large linear systems. These linear systems are solved using a Krylov subspace method. Initial guesses are constructed via a Petrov-Galerkin process applied to low-dimensional approximation subspaces derived from previous steps. Error estimates are provided, linking the accuracy of the initial guesses to the timestep size, the scheme's order, and the subspace dimension. Numerical experiments show speedups of up to two orders of magnitude over standard predictor-based methods, when those converge