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The tensor-train mimetic finite difference method for three-dimensional Maxwell’s wave propagation equations

Author

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  • Manzini, G.
  • Truong, P.M.D.
  • Vuchkov, R.
  • Alexandrov, B.

Abstract

Coupling the mimetic finite difference method with the tensor-train format results in a very effective method for low-rank numerical approximations of the solutions of the time-dependent Maxwell wave propagation equations in three dimensions. To this end, we discretize the curl operators on the primal/dual tensor product grid complex and we couple the space discretization with a staggered-in-time second-order accurate time-marching scheme. The resulting solver is accurate to the second order in time and space, and still compatible, so that the approximation of the magnetic flux field has zero discrete divergence with a discrepancy close to the machine precision level. Our approach is not limited to the second-order of accuracy. We can devise higher-order formulations in space through suitable extensions of the tensor-train stencil to compute the derivatives of the mimetic differential operators. Employing the tensor-train format improves the solver performance by orders of magnitude in terms of CPU time and memory storage. A final set of numerical experiments confirms this expectation.

Suggested Citation

  • Manzini, G. & Truong, P.M.D. & Vuchkov, R. & Alexandrov, B., 2023. "The tensor-train mimetic finite difference method for three-dimensional Maxwell’s wave propagation equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 210(C), pages 615-639.
    • Handle: RePEc:eee:matcom:v:210:y:2023:i:c:p:615-639
    DOI: 10.1016/j.matcom.2023.03.026
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    References listed on IDEAS

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    1. Vladimir Kazeev & Mustafa Khammash & Michael Nip & Christoph Schwab, 2014. "Direct Solution of the Chemical Master Equation Using Quantized Tensor Trains," PLOS Computational Biology, Public Library of Science, vol. 10(3), pages 1-19, March.
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    Cited by:

    1. Dibyendu Adak & Duc P. Truong & Gianmarco Manzini & Kim Ø. Rasmussen & Boian S. Alexandrov, 2024. "Tensor Network Space-Time Spectral Collocation Method for Time-Dependent Convection-Diffusion-Reaction Equations," Mathematics, MDPI, vol. 12(19), pages 1-20, September.

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