Tensor Network Space-Time Spectral Collocation Method for Solving the Nonlinear Convection Diffusion Equation

Adak, Dibyendu; Danis, M. Engin; Truong, Duc P.; Rasmussen, Kim Ø.; Alexandrov, Boian S.

doi:10.1007/s10915-025-02860-x

Tensor Network Space-Time Spectral Collocation Method for Solving the Nonlinear Convection Diffusion Equation

Journal Article · Fri Mar 28 00:00:00 EDT 2025 · Journal of Scientific Computing

DOI:https://doi.org/10.1007/s10915-025-02860-x· OSTI ID:2543162

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Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Spectral methods provide highly accurate numerical solutions for partial differential equations, exhibiting exponential convergence with the number of spectral nodes. Traditionally, in addressing time-dependent nonlinear problems, attention has been on low-order finite difference schemes for time discretization and spectral element schemes for spatial variables. However, our recent developments have resulted in the application of spectral methods to both space and time variables, preserving spectral convergence in both domains. Leveraging Tensor Train techniques, our approach tackles the curse of dimensionality inherent in space-time methods. Here, we extend this methodology to the nonlinear time-dependent convection-diffusion equation. Our discretization scheme exhibits a low-rank structure, facilitating translation to tensor-train (TT) format. Nevertheless, controlling the TT-rank across Newton’s iterations, needed to deal with the nonlinearity, poses a challenge, leading us to devise the “Step Truncation TT-Newton" method. We demonstrate the exponential convergence of our methods through various benchmark examples. Importantly, our scheme offers significantly reduced memory requirement compared to the full-grid scheme.

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Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE; USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 2543162

Alternate ID(s):: OSTI ID: 2570777

Report Number(s):: LA-UR--24-25458; 1573-7691; 10.1007/s10915-025-02860-x

Journal Information:: Journal of Scientific Computing, Journal Name: Journal of Scientific Computing Journal Issue: 2 Vol. 103; ISSN 1573-7691; ISSN 0885-7474

Publisher:: SpringerCopyright Statement

Country of Publication:: United States

Language:: English

References (31)

An exact solution for Burger's equation Wood, W. L. Communications in Numerical Methods in Engineering, Vol. 22, Issue 7 https://doi.org/10.1002/cnm.850	journal	January 2006
Spectral Elements for Transport-Dominated Equations Funaro, Daniele Lecture Notes in Computational Science and Engineering https://doi.org/10.1007/978-3-642-59185-3	book	April 1997
Legendre spectral collocation in space and time for PDEs Lui, S. H. Numerische Mathematik, Vol. 136, Issue 1 https://doi.org/10.1007/s00211-016-0834-x	journal	August 2016
Tensor Networks and Hierarchical Tensors for the Solution of High-Dimensional Partial Differential Equations Bachmayr, Markus; Schneider, Reinhold; Uschmajew, André Foundations of Computational Mathematics, Vol. 16, Issue 6 https://doi.org/10.1007/s10208-016-9317-9	journal	April 2016
A projector-splitting integrator for dynamical low-rank approximation Lubich, Christian; Oseledets, Ivan V. BIT Numerical Mathematics, Vol. 54, Issue 1 https://doi.org/10.1007/s10543-013-0454-0	journal	November 2013
A rank-adaptive robust integrator for dynamical low-rank approximation Ceruti, Gianluca; Kusch, Jonas; Lubich, Christian BIT Numerical Mathematics, Vol. 62, Issue 4 https://doi.org/10.1007/s10543-021-00907-7	journal	January 2022
Adaptive Integration of Nonlinear Evolution Equations on Tensor Manifolds Rodgers, Abram; Dektor, Alec; Venturi, Daniele Journal of Scientific Computing, Vol. 92, Issue 2 https://doi.org/10.1007/s10915-022-01868-x	journal	June 2022
Implicit Integration of Nonlinear Evolution Equations on Tensor Manifolds Rodgers, Abram; Venturi, Daniele Journal of Scientific Computing, Vol. 97, Issue 2 https://doi.org/10.1007/s10915-023-02352-w	journal	September 2023
Spectral collocation methods Hussaini, M. Y.; Kopriva, D. A.; Patera, A. T. Applied Numerical Mathematics, Vol. 5, Issue 3 https://doi.org/10.1016/0168-9274(89)90033-0	journal	May 1989
Jacobian-free Newton–Krylov methods: a survey of approaches and applications Knoll, D. A.; Keyes, D. E. Journal of Computational Physics, Vol. 193, Issue 2 https://doi.org/10.1016/j.jcp.2003.08.010	journal	January 2004
Tensor networks for solving the time-independent Boltzmann neutron transport equation Truong, Duc P.; Ortega, Mario I.; Boureima, Ismael Journal of Computational Physics, Vol. 507 https://doi.org/10.1016/j.jcp.2024.112943	journal	June 2024
TT-cross approximation for multidimensional arrays Oseledets, Ivan; Tyrtyshnikov, Eugene Linear Algebra and its Applications, Vol. 432, Issue 1 https://doi.org/10.1016/j.laa.2009.07.024	journal	January 2010
Collocation methods for nonlinear differential equations on low-rank manifolds Dektor, Alec Linear Algebra and its Applications, Vol. 705 https://doi.org/10.1016/j.laa.2024.11.001	journal	January 2025
The tensor-train mimetic finite difference method for three-dimensional Maxwell’s wave propagation equations Manzini, G.; Truong, P. M. D.; Vuchkov, R. Mathematics and Computers in Simulation, Vol. 210 https://doi.org/10.1016/j.matcom.2023.03.026	journal	August 2023
Quantized tensor networks for solving the Vlasov–Maxwell equations Ye, Erika; Loureiro, Nuno F. Journal of Plasma Physics, Vol. 90, Issue 3 https://doi.org/10.1017/S0022377824000503	journal	June 2024
Quantum-inspired framework for computational fluid dynamics Peddinti, Raghavendra Dheeraj; Pisoni, Stefano; Marini, Alessandro Communications Physics, Vol. 7, Issue 1 https://doi.org/10.1038/s42005-024-01623-8	journal	April 2024
Solving the time-independent Schrödinger equation for chains of coupled excitons and phonons using tensor trains Gelß, Patrick; Klein, Rupert; Matera, Sebastian The Journal of Chemical Physics, Vol. 156, Issue 2 https://doi.org/10.1063/5.0074948	journal	January 2022
CUR matrix decompositions for improved data analysis Mahoney, Michael W.; Drineas, Petros Proceedings of the National Academy of Sciences, Vol. 106, Issue 3 https://doi.org/10.1073/pnas.0803205106	journal	January 2009
Guaranteed a posteriori error bounds for low-rank tensor approximate solutions Dolgov, Sergey; Vejchodský, Tomáš IMA Journal of Numerical Analysis, Vol. 41, Issue 2 https://doi.org/10.1093/imanum/draa010	journal	June 2020
Fast adaptive interpolation of multi-dimensional arrays in tensor train format Savostyanov, Dmitry; Oseledets, Ivan The 2011 International Workshop on Multidimensional (nD) Systems https://doi.org/10.1109/nDS.2011.6076873	conference	September 2011
Dynamic Programming Bellman, R. Science, Vol. 153, Issue 3731 https://doi.org/10.1126/science.153.3731.34	journal	July 1966
Tensor Decompositions and Applications Kolda, Tamara G.; Bader, Brett W. SIAM Review, Vol. 51, Issue 3 https://doi.org/10.1137/07070111X	journal	August 2009
Tensor-Train Decomposition Oseledets, I. V. SIAM Journal on Scientific Computing, Vol. 33, Issue 5 https://doi.org/10.1137/090752286	journal	January 2011
Approximation of $2^d\times2^d$ Matrices Using Tensor Decomposition Oseledets, I. V. SIAM Journal on Matrix Analysis and Applications, Vol. 31, Issue 4 https://doi.org/10.1137/090757861	journal	January 2010
Iterative Methods for Linear and Nonlinear Equations Kelley, C. T. Frontiers in Applied Mathematics https://doi.org/10.1137/1.9781611970944	book	January 1995
The Alternating Linear Scheme for Tensor Optimization in the Tensor Train Format Holtz, Sebastian; Rohwedder, Thorsten; Schneider, Reinhold SIAM Journal on Scientific Computing, Vol. 34, Issue 2 https://doi.org/10.1137/100818893	journal	January 2012
Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions Dolgov, Sergey V.; Savostyanov, Dmitry V. SIAM Journal on Scientific Computing, Vol. 36, Issue 5 https://doi.org/10.1137/140953289	journal	January 2014
Regularization of Poisson--Boltzmann Type Equations with Singular Source Terms Using the Range-Separated Tensor Format Benner, Peter; Khoromskaia, Venera; Khoromskij, Boris SIAM Journal on Scientific Computing, Vol. 43, Issue 1 https://doi.org/10.1137/19M1281435	journal	January 2021
QTT approximation of elliptic solution operators in higher dimensions Khoromskij, B. N.; Oseledets, I. V. Russian Journal of Numerical Analysis and Mathematical Modelling, Vol. 26, Issue 3 https://doi.org/10.1515/rjnamm.2011.017	journal	January 2011
Tensor-Train TENO Scheme for Compressible Flows Danis, Engin; Truong, Duc; Boureima, Ismael AIAA SCITECH 2025 Forum https://doi.org/10.2514/6.2025-0304	conference	January 2025
Tensor Network Space-Time Spectral Collocation Method for Time-Dependent Convection-Diffusion-Reaction Equations Adak, Dibyendu; Truong, Duc P.; Manzini, Gianmarco Mathematics, Vol. 12, Issue 19 https://doi.org/10.3390/math12192988	journal	September 2024

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