Computing Linear Combinations of $\phi$-Function Actions for Exponential Integrators

Abstract

We propose a matrix-free algorithm for evaluating linear combinations of $\phi$-function actions, $w_i:={\sum }_{j=0}^p{\alpha }_i^j{\phi }_j(t_{i}A)v_j$ for $i=1:r$, arising in exponential integrators. The method combines the scaling and recovering method with a truncated Taylor series, choosing a spectral shift and a scaling parameter by minimizing a power-based objective of the shifted operator. Accuracy is user-controlled and ultimately limited by the working precision. The algorithm decouples the stage abscissae $t_i$ from the polynomial weights ${\alpha }_i^j$, and a block variant enables simultaneous evaluation of $\{w_i{\}}_{i=1}^r$. Across standard benchmarks, including stiff and highly nonnormal matrices, the algorithm attains near-machine accuracy (IEEE double precision in our tests) for small step sizes and maintains reliable accuracy for larger steps where several existing Krylov-based algorithms deteriorate, providing a favorable balance of reliability and computational cost.

17 pages, 3 figures