A high-order exponential integrator for nonlinear parabolic equations with nonsmooth initial data

TL;DR

This paper introduces a variable stepsize exponential multistep integrator with contour integral approximation for solving semilinear parabolic equations with nonsmooth initial data, achieving high-order convergence.

Contribution

It proposes a novel high-order exponential integrator that effectively handles nonsmooth initial data with theoretical convergence guarantees.

Findings

Achieves high-order convergence in maximum norm

Handles nonsmooth initial data effectively

Validated by numerical example

Abstract

A variable stepsize exponential multistep integrator, with contour integral approximation of the operator-valued exponential functions, is proposed for solving semilinear parabolic equations with nonsmooth initial data. By this approach, the exponential k-step method would have $k$th-order convergence in approximating a mild solution, possibly nonsmooth at the initial time. In consistency with the theoretical analysis, a numerical example shows that the method can achieve high-order convergence in the maximum norm for semilinear parabolic equations with discontinuous initial data.