On the implementation of exponential methods for semilinear parabolic equations

TL;DR

This paper introduces a new algorithm for implementing exponential methods in solving semilinear parabolic equations, utilizing quadrature formulas based on numerical Laplace transform inversion for efficient operator evaluation.

Contribution

A novel algorithm for exponential method implementation using quadrature formulas with exponential convergence based on Laplace transform inversion.

Findings

Quadrature formula converges like $O(e^{-cK/ ext{ln} K})$ for operator evaluation.

Enhanced evaluation of scalar mappings with quadrature convergence like $O(e^{-cK})$.

Numerical tests demonstrate the algorithm's effectiveness.

Abstract

The time integration of semilinear parabolic problems by exponential methods of different kinds is considered. A new algorithm for the implementation of these methods is proposed. The algorithm evaluates the operators required by the exponential methods by means of a quadrature formula that converges like $O(e^{-cK/\ln K})$, with $K$ the number of quadrature nodes. The algorithm allows also the evaluation of the associated scalar mappings and in this case the quadrature converges like $O(e^{-cK})$. The technique is based on the numerical inversion of sectorial Laplace transforms. Several numerical illustrations are provided to test the algorithm.