Analysis of exponential splitting methods for inhomogeneous parabolic equations

TL;DR

This paper investigates the convergence behavior of exponential splitting methods for inhomogeneous parabolic equations, revealing conditions for full-order convergence and explaining observed order reductions through theoretical analysis and numerical experiments.

Contribution

It provides sharp convergence results for exponential Lie and Strang splitting methods applied to inhomogeneous parabolic equations, including conditions for full-order convergence and explanations for order reduction.

Findings

Full-order convergence for homogeneous boundary conditions.

Order reduction to 1.25 for Strang splitting with inhomogeneity.

Numerical experiments confirm theoretical convergence rates.

Abstract

We analyze the convergence of the exponential Lie and exponential Strang splitting applied to inhomogeneous second-order parabolic equations with Dirichlet boundary conditions. A recent result on the smoothing properties of these methods allows us to prove sharp convergence results in the case of homogeneous Dirichlet boundary conditions. When no source term is present and natural regularity assumptions are imposed on the initial value, we show full-order convergence of both methods. For inhomogeneous equations, we prove full-order convergence for the exponential Lie splitting, whereas order reduction to 1.25 for the exponential Strang splitting. Furthermore, we give sufficient conditions on the inhomogeneity for full-order convergence of both methods. Moreover our theoretical convergence results explain the severe order reduction to 0.25 of splitting methods applied to problems involving inhomogeneous Dirichlet boundary conditions. We include numerical experiments to underline the sharpness of our theoretical convergence results.