[Paper Review] Computation of Maxwell's equations on Manifold using DEC

Zheng Xie, Yujie Ma|ArXiv.org|Aug 31, 2009

Electromagnetic Simulation and Numerical Methods17 references4 citations

TL;DR

This paper presents a novel discrete exterior calculus (DEC) framework for solving Maxwell’s equations on curved manifolds using a spacetime prism lattice, enabling stable and accurate simulation of electromagnetic wave propagation in 2D and 3D geometries. The method preserves key geometric structures, supports unstructured grids, and achieves first-order temporal and spatial accuracy, with second-order accuracy on structured grids.

ABSTRACT

In this paper, the method of discrete exterior calculus for numerically solving Maxwell's equations in space manifold and the time is discussed, which is a kind of lattice gauge theory. The analysis of its stable condition and error is also accomplished. This algorithm has been implemented on C++ plateform for simulating TE/M waves in vacuum.

Motivation & Objective

• To develop a geometric numerical method for solving Maxwell’s equations on general 2D and 3D manifolds using discrete exterior calculus (DEC).
• To extend the Yee scheme to spacetime prism lattices, enabling unstructured spatial grids and variable time steps.
• To ensure stability and convergence of the scheme through rigorous analysis of truncation error and eigenvalue conditions.
• To implement and validate the method on C++ for simulating TE/TM wave modes in vacuum on complex surfaces and 3D manifolds.

Proposed method

• Discretizes spacetime as a product of 1D time segments and 2D/3D spatial triangulations or tetrahedrons, forming a prism lattice.
• Defines a discrete Lorentz metric by combining spatial Riemann metrics (edge lengths, face areas, volumes) with a negative time metric.
• Uses circumcentric dual cells to define discrete Hodge star operators that scale primal and dual cell volumes.
• Implements discrete exterior derivative $d$ as the transpose of the incidence matrix between k- and (k+1)-simplices.
• Derives discrete Maxwell’s equations via variational principles on discrete 1-forms, yielding source and continuity equations.
• Splits the system into explicit iterative schemes for TE and TM modes, using local geometric data like edge lengths and face areas.

Experimental results

Research questions

• RQ1 How can Maxwell’s equations be consistently discretized on unstructured spatial manifolds while preserving their geometric and topological structure?
• RQ2 What are the stability conditions for a DEC-based time-domain scheme with variable time steps on a spacetime prism lattice?
• RQ3 What is the convergence and accuracy order of the proposed DEC scheme for electromagnetic wave propagation?
• RQ4 Can the method simulate complex wave phenomena such as interference and propagation on curved surfaces and 3D manifolds?
• RQ5 How does the scheme perform on non-rectangular grids, and what is its accuracy on structured versus unstructured grids?

Key findings

• The DEC scheme satisfies the discrete Bianchi identity $dF = 0$ automatically, ensuring local conservation of the electromagnetic field curvature.
• The method achieves first-order temporal and spatial accuracy in general unstructured grids, with second-order accuracy on rectangular grids when time and space steps are equivalent.
• Stability is guaranteed under a CFL-type condition derived from temporal and spatial eigenvalue analysis: $\Delta t \leq \min_{P_{123}} \left( \frac{1}{c} \sqrt{ \frac{2P_{123}}{ \left( \frac{l_{23}}{l_{A0}} + \frac{l_{34}}{l_{B0}} + \frac{l_{45}}{l_{C0}} \right) } } \right)$.
• The scheme is consistent with the continuous Maxwell’s equations, with truncation error $O((\Delta t)^2 + \Delta t |*e|)$, ensuring convergence via the Lax equivalence theorem.
• Numerical implementation in C++ successfully simulates Gaussian pulse propagation and interference on surfaces such as the Stanford bunny and a sphere.
• The method supports arbitrary triangulations, including acute triangles, rectangles, and regular polygons, enabling flexible simulation on complex geometries.