Exponential Rosenbrock methods without order reduction when integrating nonlinear initial value problems

TL;DR

This paper introduces a technique to prevent order reduction in explicit exponential Rosenbrock methods for reaction-diffusion problems, enhancing efficiency without requiring stiff order conditions or specific boundary value constraints.

Contribution

It presents a novel technique applicable to any Rosenbrock method that avoids order reduction in reaction-diffusion problems, regardless of boundary conditions.

Findings

The technique effectively prevents order reduction in numerical integration.

Numerical experiments show significant efficiency gains over standard methods.

Theoretical analysis confirms the method's accuracy and robustness.

Abstract

A technique is described in this paper to avoid order reduction when integrating reaction-diffusion initial boundary value problems with explicit exponential Rosenbrock methods. The technique is valid for any Rosenbrock method, without having to impose any stiff order conditions, and for general time-dependent boundary values. An analysis on the global error is thoroughly performed and some numerical experiments are shown which corroborate the theoretical results, and in which a big gain in efficiency with respect to applying the standard method of lines can be observed.