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

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.