Rosenbrock-Krylov Methods for Large Systems of Differential Equations

TL;DR

This paper introduces Rosenbrock-Krylov methods, a new class of integrators for large-scale ODEs and PDEs, combining Krylov space solutions with Rosenbrock schemes for improved efficiency and accuracy.

Contribution

It develops a unified Rosenbrock-Krylov framework that integrates Krylov space approximation into the order condition theory, reducing basis size requirements and simplifying error monitoring.

Findings

Favorable numerical performance compared to existing Rosenbrock methods

Small, order-dependent Krylov subspace size

No need for error monitoring in linear system solutions

Abstract

This paper develops a new class of Rosenbrock-type integrators based on a Krylov space solution of the linear systems. The new family, called Rosenbrock-Krylov (Rosenbrock-K), is well suited for solving large scale systems of ODEs or semi-discrete PDEs. The time discretization and the Krylov space approximation are treated as a single computational process, and the Krylov space properties are an integral part of the new Rosenbrock-K order condition theory developed herein. Consequently, Rosenbrock-K methods require a small number of basis vectors determined solely by the temporal order of accuracy. The subspace size is independent of the ODE under consideration, and there is no need to monitor the errors in linear system solutions at each stage. Numerical results show favorable properties of Rosenbrock-K methods when compared to current Rosenbrock and Rosenbrock-W schemes.