Space-Time Spectral Collocation Tensor-Network Approach for Maxwell's Equations
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909967757344768 |
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| author | Adak, Dibyendu Chinomona, Rujeko Truong, Duc P. Korobkin, Oleg Rasmussen, Kim Ø. Alexandrov, Boian S. |
| author_facet | Adak, Dibyendu Chinomona, Rujeko Truong, Duc P. Korobkin, Oleg Rasmussen, Kim Ø. Alexandrov, Boian S. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15631 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Space-Time Spectral Collocation Tensor-Network Approach for Maxwell's Equations Adak, Dibyendu Chinomona, Rujeko Truong, Duc P. Korobkin, Oleg Rasmussen, Kim Ø. Alexandrov, Boian S. Numerical Analysis 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. |
| title | Space-Time Spectral Collocation Tensor-Network Approach for Maxwell's Equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2512.15631 |