A Krylov subspace algorithm for evaluating the phi-functions appearing in exponential integrators
This work provides a more efficient algorithm for a core computational task in exponential integrators, benefiting researchers solving large systems of ODEs.
The paper develops an adaptive Krylov subspace algorithm for computing the action of phi-functions on vectors, which is the main computational bottleneck in exponential integrators. Numerical experiments show the algorithm is often significantly more efficient than state-of-the-art methods.
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.