Convergence analysis of the Magnus-Rosenbrock type method for the finite element discretization of semilinear non-autonomous parabolic PDEs with nonsmooth initial data

TL;DR

This paper introduces a new Magnus-Rosenbrock numerical scheme for semilinear non-autonomous parabolic PDEs with nonsmooth initial data, demonstrating stability and convergence properties tailored to reactive dominated transport equations.

Contribution

The paper develops a novel explicit Magnus-Rosenbrock method that is stable, efficient, and convergent for challenging PDEs with nonsmooth initial data, extending existing numerical analysis.

Findings

Achieves convergence orders of O(h^2 + Δt^{2-ε}) under certain regularity conditions

Proves stability and convergence of the proposed scheme for reactive dominated transport equations

Numerical simulations confirm theoretical convergence rates

Abstract

This paper aims to investigate a full numerical approximation of non-autonomous semilnear parabolic partial differential equations (PDEs) with nonsmooth initial data. Our main interest is on such PDEs where the nonlinear part is stronger than the linear part, also called reactive dominated transport equations. For such equations, many classical numerical methods lose their stability properties. We perform the space and time discretizations respectively by the finite element method and an exponential integrator. We obtain a novel explicit, stable and efficient scheme for such problems called Magnus-Rosenbrock method. We prove the convergence of the fully discrete scheme toward the exact solution. The result shows how the convergence orders in both space and time depend on the regularity of the initial data. In particular, when the initial data belongs to the domain of the family of the linear operator, we achieve convergence orders $\mathcal{O}\left(h^{2}+\Delta t^{2-\epsilon}\right)$, for an arbitrarily small $\epsilon>0$. Numerical simulations to illustrate our theoretical result are provided.