Why Improving the Accuracy of Exponential Integrators Can Decrease Their Computational Cost?

Begoña Cano,Nuria Reguera

doi:10.3390/math9091008

Abstract

In previous papers, a technique has been suggested to avoid order reduction when integrating initial boundary value problems with several kinds of exponential methods. The technique implies in principle to calculate additional terms at each step from those already necessary without avoiding order reduction. The aim of the present paper is to explain the surprising result that, many times, in spite of having to calculate more terms at each step, the computational cost of doing it through Krylov methods decreases instead of increases. This is very interesting since, in that way, the methods improve not only in terms of accuracy, but also in terms of computational cost.

Highlights

Exponential methods have become a valuable tool to integrate initial boundary value problems due to the recent development of techniques which allow us to calculate exponential-type functions in a more efficient way
Order reduction is observed with respect to their classical counterparts unless more stringent conditions of annihilation on the boundary or periodic boundary conditions are considered
What we have observed in some numerical experiments with the techniques in those papers is that, when avoiding order reduction, it does happen that the error diminishes more quickly with the stepsize, and that, for a same moderate stepsize, the error is smaller and, what is more impressive, that the computational cost for a same stepsize is smaller

Summary

Introduction

Exponential methods have become a valuable tool to integrate initial boundary value problems due to the recent development of techniques which allow us to calculate exponential-type functions in a more efficient way. What we have observed in some numerical experiments with the techniques in those papers is that, when avoiding order reduction, it does happen that the error diminishes more quickly with the stepsize, and that, for a same moderate stepsize, the error is smaller and, what is more impressive, that the computational cost for a same stepsize is smaller This is striking because increasing the order of the method implies calculating more terms at each step. It is analysed how the subroutine works when the technique to avoid order reduction is applied in the naive way.

Preliminaries

An Example as a Motivation

Standard Krylov Method

Application to Our Problem

Purely Rational Krylov Method

General Conclusions