Analysis of a Peaceman-Rachford ADI scheme for Maxwell equations in heterogeneous media

TL;DR

This paper rigorously analyzes the Peaceman-Rachford ADI scheme for Maxwell equations in heterogeneous media, revealing a reduced convergence rate due to material discontinuities and confirming the theoretical results with numerical experiments.

Contribution

It provides the first rigorous error analysis of the ADI scheme for Maxwell equations in heterogeneous media, accounting for solution irregularities.

Findings

Proven convergence rate is half an order lower than classical expectations.

Error bounds depend only on initial data and material parameters, not on solution regularity.

Numerical experiments confirm the theoretical convergence rate as optimal.

Abstract

The Peaceman-Rachford alternating direction implicit (ADI) scheme for linear time-dependent Maxwell equations is analyzed on a heterogeneous cuboid. Due to discontinuities of the material parameters, the solution of the Maxwell equations is less than $H^2$-regular in space. For the ADI scheme, we prove a rigorous time-discrete error bound with a convergence rate that is half an order lower than the classical one. Our statement imposes only assumptions on the initial data and the material parameters, but not on the solution. To establish this result, we analyze the regularity of the Maxwell equations in detail in an appropriate functional analytical framework. The theoretical findings are complemented by a numerical experiment indicating that the proven convergence rate is indeed observable and optimal.