The Hermite-Taylor Correction Function Method for Embedded Boundary and Maxwell's Interface Problems

TL;DR

This paper introduces a high-order Hermite-Taylor correction function method for Maxwell's equations that effectively handles embedded boundary and interface conditions, improving stability and flexibility in complex electromagnetic simulations.

Contribution

The paper develops a novel correction function approach integrated with Hermite-Taylor methods to accurately enforce boundary and interface conditions in Maxwell's equations.

Findings

Enables stable enforcement of boundary conditions with relaxed time-step restrictions.

Handles discontinuous solutions at interfaces effectively.

Flexible approach adaptable to other hyperbolic systems.

Abstract

We propose a novel Hermite-Taylor correction function method to handle embedded boundary and interface conditions for Maxwell's equations. The Hermite-Taylor method evolves the electromagnetic fields and their derivatives through order $m$ in each Cartesian coordinate. This makes the development of a systematic approach to enforce boundary and interface conditions difficult. Here we use the correction function method to update the numerical solution where the Hermite-Taylor method cannot be applied directly. Time derivatives of boundary and interface conditions, converted into spatial derivatives, are enforced to obtain a stable method and relax the time-step size restriction of the Hermite-Taylor correction function method. The proposed high-order method offers a flexible systematic approach to handle embedded boundary and interface problems, including problems with discontinuous solutions at the interface. This method is also easily adaptable to other first order hyperbolic systems.