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

TL;DR

This paper introduces a tensor network spectral collocation method for nonlinear convection-diffusion equations, achieving spectral accuracy in space and time with reduced memory use via tensor train techniques.

Contribution

The paper extends spectral methods to both space and time for nonlinear PDEs using tensor train formats, introducing the Step Truncation TT-Newton method for efficient nonlinear solving.

Findings

Exponential convergence demonstrated on benchmarks

Significantly reduced memory compared to full-grid schemes

Effective control of TT-rank during iterations

Abstract

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.