Space-Time Spectral Collocation Tensor-Network Approach for Maxwell's Equations

TL;DR

This paper introduces a novel space-time spectral collocation method combined with tensor-network techniques to efficiently solve three-dimensional Maxwell's equations, achieving spectral accuracy and linear complexity in large-scale problems.

Contribution

The work develops a new space-time spectral collocation approach integrated with tensor-train formats for Maxwell's equations, enabling efficient high-accuracy solutions without separability assumptions.

Findings

Spectral convergence for electric and magnetic fields.

Linear complexity in the number of grid points.

Effective handling of large, structured linear systems.

Abstract

In this work, we develop a space--time Chebyshev spectral collocation method for three-dimensional Maxwell's equations and combine it with tensor-network techniques in Tensor-Train (TT) format. Under constant material parameters, the Maxwell system is reduced to a vector wave equation for the electric field, which we discretize globally in space and time using a staggered spectral collocation scheme. The staggered polynomial spaces are designed so that the discrete curl and divergence operators preserve the divergence-free constraint on the magnetic field. The magnetic field is then recovered in a space--time post-processing step via a discrete version of Faraday's law. The global space--time formulation yields a large but highly structured linear system, which we approximate in low-rank TT-format directly from the operator and data, without assuming that the forcing is separable in space and time. We derive condition-number bounds for the resulting operator and prove spectral convergence estimates for both the electric and magnetic fields. Numerical experiments for three-dimensional electromagnetic test problems confirm the theoretical convergence rates and show that the TT-based solver maintains accuracy with approximately linear complexity in the number of grid points in space and time.