Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Dibyendu Adak, Rujeko Chinomona, Duc P. Truong, Oleg Korobkin, Kim Ø. Rasmussen, Boian S. Alexandrov

TL;DR

This work develops a space-time Chebyshev spectral collocation method for three-dimensional Maxwell's equations with constant coefficients and couples it to tensor-train (TT) representations to overcome the curse of dimensionality. A staggered space-time discretization preserves the divergence-free constraint for the magnetic field, while a discrete Faraday post-processing recovers B from E with spectral accuracy. The authors establish condition-number bounds and exponential convergence for both fields, and demonstrate, through TT-based numerical experiments, that the method achieves exponential convergence with near-linear complexity in space-time degrees of freedom and massive memory savings compared to full-grid methods. The TT framework, together with TT-cross interpolation and TT-matrix representations, enables solving the global space-time system efficiently, enabling high-resolution Maxwell simulations that were previously intractable. Overall, the approach offers a scalable, high-accuracy toolkit for large-scale electromagnetic simulations with potential impact on optics, RF design, and computational electromagnetics at scale.

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.

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

TL;DR

This work develops a space-time Chebyshev spectral collocation method for three-dimensional Maxwell's equations with constant coefficients and couples it to tensor-train (TT) representations to overcome the curse of dimensionality. A staggered space-time discretization preserves the divergence-free constraint for the magnetic field, while a discrete Faraday post-processing recovers B from E with spectral accuracy. The authors establish condition-number bounds and exponential convergence for both fields, and demonstrate, through TT-based numerical experiments, that the method achieves exponential convergence with near-linear complexity in space-time degrees of freedom and massive memory savings compared to full-grid methods. The TT framework, together with TT-cross interpolation and TT-matrix representations, enables solving the global space-time system efficiently, enabling high-resolution Maxwell simulations that were previously intractable. Overall, the approach offers a scalable, high-accuracy toolkit for large-scale electromagnetic simulations with potential impact on optics, RF design, and computational electromagnetics at scale.

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.

Paper Structure

This paper contains 22 sections, 7 theorems, 87 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Mathematical model and discretization
  3. Space--time Chebyshev spectral collocation
  4. Staggered space--time discrete spaces
  5. Post-processing to recover the magnetic field
  6. Wave equation in the space--time spectral collocation scheme
  7. Recovering the magnetic field
  8. Condition number estimates and spectral convergence
  9. Condition number estimates
  10. Spectral convergence
  11. Tensor networks
  12. Tensor train representation
  13. Linear operators in TT-matrix format
  14. TT cross interpolation
  15. Tensorization of the electric field
  16. ...and 7 more sections

Key Result

Lemma 4.1

Let $N \geq 1$, and $\lambda \in \Lambda (\langle \mathbf{S} \rangle)$. Then where $C$ is a positive constant independent of $N$.

Figures (5)

  • Figure 1: TT decomposition of a 4D tensor $\mathcal{X}$, with TT-rank $\mathbf{r} = [r_1,r_2,r_3]$ and approximation error $\varepsilon$, in accordance with Eq. \ref{['eqn:TT_def_element']}.
  • Figure 2: Representation of a linear matrix $\mathbf{A}_{Lap}$ in the TT-matrix format. First, we reshape the operator matrix $\mathbf{A}_{Lap}$ and permute its indices to create the tensor $\mathcal{A}$. Then, we factorize the tensor in the tensor-train matrix format according to Eq. \ref{['eqn:TT-matrix-componentwise']} to obtain $\mathcal{A}^{\text{TT}{}}$.
  • Figure 3: Test Case 1, upper panels: $L_2$-errors in the computation of the electric and magnetic fields (upper left), and divergence constraints: $\|\mathop{\mathrm{div}}\nolimits \mathbf{E} - \rho\|$, $\| \mathop{\mathrm{div}}\nolimits \mathbf{B} \|$ (upper right). Bottom panel: elapsed time for the full grid calculation vs TT representation.
  • Figure 4: Test Case 2, upper panels: $L_2$-errors of electric and magnetic fields (upper left), and divergence constraints: $\|\mathop{\mathrm{div}}\nolimits \mathbf{E} - \rho \|$, $\| \mathop{\mathrm{div}}\nolimits \mathbf{B} \|$ (upper right). Bottom panel: elapsed time for full grid calculation vs TT representation.
  • Figure 5: Test Case 3, upper panels: $L_2$-error in the computation of the electric and magnetic fields (upper left), and divergence constraint $\mathop{\mathrm{div}}\nolimits \mathbf{B} = 0$ (upper right). Bottom panel: elapsed time for full-grid calculation vs. TT representation.

Theorems & Definitions (12)

  • Lemma 4.2
  • Lemma 4.3
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • proof
  • Theorem 4.4
  • proof
  • ...and 2 more