EPIRK-W and EPIRK-K time discretization methods

TL;DR

This paper introduces two new families of exponential integrators, EPIRK-W and EPIRK-K, that efficiently incorporate Jacobian approximations to reduce computational costs while maintaining accuracy.

Contribution

The paper extends the EPIRK family to include inexact Jacobians and develops new methods with classical order conditions, improving efficiency of exponential integrators.

Findings

EPIRK-W and EPIRK-K methods are computationally advantageous

Developed practical third- and fourth-order methods

Numerical experiments show improved efficiency over existing integrators

Abstract

Exponential integrators are special time discretization methods where the traditional linear system solves used by implicit schemes are replaced with computing the action of matrix exponential-like functions on a vector. A very general formulation of exponential integrators is offered by the Exponential Propagation Iterative methods of Runge-Kutta type (EPIRK) family of schemes. The use of Jacobian approximations is an important strategy to drastically reduce the overall computational costs of implicit schemes while maintaining the quality of their solutions. This paper extends the EPIRK class to allow the use of inexact Jacobians as arguments of the matrix exponential-like functions. Specifically, we develop two new families of methods: EPIRK-W integrators that can accommodate any approximation of the Jacobian, and EPIRK-K integrators that rely on a specific Krylov-subspace projection of the exact Jacobian. Classical order conditions theories are constructed for these families. A practical EPIRK-W method of order three and an EPIRK-K method of order four are developed. Numerical experiments indicate that the methods proposed herein are computationally favorable when compared to existing exponential integrators.