Block Krylov subspace exact time integration of linear ODE systems. Part 1: algorithm description

TL;DR

This paper introduces a time-exact method for solving linear ODE systems using Krylov subspaces, combining polynomial approximation of source terms with residual-based Krylov methods for high accuracy.

Contribution

It presents a novel two-stage approach that achieves time-exact solutions by integrating polynomial approximation and residual Krylov subspace techniques.

Findings

Method achieves high accuracy limited only by approximation and Krylov errors

Polynomial approximation of source term g(t) is highly accurate

Residual Krylov method ensures precise solution computation

Abstract

We propose a time-exact Krylov-subspace-based method for solving linear ODE (ordinary differential equation) systems of the form $y'=-Ay + g(t)$, where $y(t)$ is the unknown function. The method consists of two stages. The first stage is an accurate polynomial approximation of the source term $g(t)$, constructed with the help of the truncated SVD (singular value decomposition). The second stage is a special residual-based block Krylov subspace method. The accuracy of the method is only restricted by the accuracy of the polynomial approximation and by the error of the block Krylov process. Since both errors can, in principle, be made arbitrarily small, this yields, at some costs, a time-exact method.