A higher-order dual cell method for time-domain Maxwell equations

TL;DR

This paper introduces a high-order dual cell method for 3D time-domain Maxwell equations, achieving spectral accuracy, energy preservation, and efficiency on unstructured meshes.

Contribution

It extends a 2D high-order dual cell method to 3D, using tensor-product polynomial spaces and compatible discretizations for Maxwell equations.

Findings

Achieves arbitrary-order convergence and spectral correctness.

Maintains energy identity and spurious mode avoidance.

Demonstrates high accuracy and efficiency on unstructured tetrahedral meshes.

Abstract

We present a higher-order extension of the dual cell method for the time-domain Maxwell equations in three spatial dimensions. The approach builds upon a variational reinterpretation of the Finite Integration Technique on dual meshes and generalises a previously developed two-dimensional high-order formulation. The electric and magnetic fields are discretised on mutually dual barycentric grids using curl-conforming polynomial spaces constructed via tensor-product Gauss--Radau interpolation. The resulting semi-discrete formulation yields block-diagonal mass matrices and sparse discrete curl operators, enabling explicit time integration while preserving a discrete energy identity. Special attention is devoted to the construction of compatible approximation spaces on the three-dimensional primal and dual meshes, the reference-to-physical element mappings, and the preservation of tangential continuity. We show that the method achieves arbitrary-order convergence, avoids spurious modes, and maintains optimal sparsity properties. Numerical experiments confirm spectral correctness, high-order accuracy, and computational efficiency on unstructured tetrahedral meshes.