Krylov subspace residual and restarting for certain second order differential equations

TL;DR

This paper introduces efficient algorithms for time integration of large oscillatory second order ODEs using Krylov subspace residuals and restarting techniques, extending methods from first order ODEs.

Contribution

It extends residual-time restarting Krylov subspace methods to second order ODEs with oscillatory solutions, reducing computational costs with new theoretical analysis.

Findings

Algorithms demonstrate improved efficiency in numerical experiments.

Residual convergence analyzed via Faber and Chebyshev series.

Restarting reduces computational cost in the Gautschi cosine scheme.

Abstract

We propose algorithms for efficient time integration of large systems of oscillatory second order ordinary differential equations (ODEs) whose solution can be expressed in terms of trigonometric matrix functions. Our algorithms are based on a residual notion for second order ODEs, which allows to extend the ``residual-time restarting'' Krylov subspace framework -- which was recently introduced for exponential and $\varphi$-functions occurring in time integration of first order ODEs -- to our setting. We then show that the computational cost can be further reduced in many cases by using our restarting in the Gautschi cosine scheme. We analyze residual convergence in terms of Faber and Chebyshev series and supplement these theoretical results by numerical experiments illustrating the efficiency of the proposed methods.