A Krylov subspace algorithm for evaluating the phi-functions appearing in exponential integrators

TL;DR

The paper introduces an adaptive Krylov subspace algorithm for efficiently computing matrix functions like phi-functions, crucial for exponential integrators in solving large linear ODE systems, outperforming existing methods.

Contribution

It develops a fully adaptive Krylov subspace algorithm combining Arnoldi or Lanczos iteration with time-stepping, improving efficiency in computing matrix functions for exponential integrators.

Findings

The algorithm is often significantly more efficient than current methods.

It adaptively varies time step and Krylov subspace size for accuracy.

Implemented in Matlab as the phipm function with practical instructions.

Abstract

We develop an algorithm for computing the solution of a large system of linear ordinary differential equations (ODEs) with polynomial inhomogeneity. This is equivalent to computing the action of a certain matrix function on the vector representing the initial condition. The matrix function is a linear combination of the matrix exponential and other functions related to the exponential (the so-called phi-functions). Such computations are the major computational burden in the implementation of exponential integrators, which can solve general ODEs. Our approach is to compute the action of the matrix function by constructing a Krylov subspace using Arnoldi or Lanczos iteration and projecting the function on this subspace. This is combined with time-stepping to prevent the Krylov subspace from growing too large. The algorithm is fully adaptive: it varies both the size of the time steps and the dimension of the Krylov subspace to reach the required accuracy. We implement this algorithm in the Matlab function phipm and we give instructions on how to obtain and use this function. Various numerical experiments show that the phipm function is often significantly more efficient than the state-of-the-art.