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

TL;DR

The paper addresses high-order numerical treatment of Maxwell's equations on domains with embedded boundaries and interfaces, where classical Hermite-Taylor methods struggle to enforce boundary/interface conditions. It introduces a Hermite-Taylor correction function method (CFM) that updates the solution at CF nodes by solving local space-time minimization problems, with time derivatives converted to spatial derivatives to improve stability. The method achieves high-order accuracy via correction-function polynomials defined on local patches, supported by a well-posed quadratic optimization with precomputable matrices, and demonstrated stability and convergence in 1D and 2D tests including discontinuous solutions at interfaces. The approach is adaptable to other first-order hyperbolic systems and balances accuracy, stability, and parallelizability, though matrix conditioning limits the feasible order in multi-D cases.

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.

Figures (33)

• Figure 1: Geometry of a domain $\Omega$. The domain consists of two materials.
• Figure 2: Illustration of the domain of integration $S_0\times [t_{n-1},t_n]$ of $\mathcal{G}_0^n$. The primal CF and Hermite nodes are respectively represented by green squares and black squares while the dual Hermite nodes are represented by blue circles. The CFM seeks the information located at $(x_1,t_{n})$ which is enclosed by the red circle. The space-time local patch $S_0\times [t_{n-1},t_n]$ is denoted by a magenta box.
• Figure 3: Illustration of the domain of integration $[t_{n-1},t_n]$ at $x_\ell$ of $\mathcal{B}_0^n$. The primal CF and Hermite nodes are respectively represented by green squares and black squares while the dual Hermite nodes are represented by blue circles. The CFM seeks the information located at $(x_1,t_{n})$ which is enclosed by the red circle. The intersection between the boundary and the local patch, that is the line connecting $(x_\ell,t_{n-1})$ to $(x_\ell,t_n)$, is denoted by a dashed purple line.
• Figure 4: Illustration of the domains of integration $S_{0,p}^\mathcal{H} \times[t_{n-1/2},t_n]$ and $S_{0,d}^\mathcal{H} \times[t_{n-1},t_{n-1/2}]$ of $\mathcal{H}_0^n$. The primal CF and Hermite nodes are respectively represented by green squares and black squares while the dual Hermite nodes are represented by blue circles. The CFM seeks the information located at $(x_1,t_{n})$ which is enclosed by the red circle. The domains $S_{0,p}^\mathcal{H} \times [t_{n-1/2},t_n]$ and $S_{0,d}^\mathcal{H} \times [t_{n-1},t_{n-1/2}]$, where we enforce the correction functions to match the Hermite-Taylor polynomials, are denoted by blue boxes.
• Figure 5: Illustration of the domain of integration $S_0\times [t_{n-1},t_n]$ of $\mathcal{G}_0^{+,n}$ and $\mathcal{G}_0^{-,n}$. The primal CF and Hermite nodes are respectively represented by green squares and black squares while the dual CF and Hermite nodes are represented by green circles and blue circles. The CFM seeks the information located at $(x_{i+1},t_{n})$ which is enclosed by the red circle. The space-time local patch $S_0\times [t_{n-1},t_n]$ is denoted by a magenta box.

Theorems & Definitions (8)

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