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

TL;DR

This work addresses solving nonlinear time-dependent convection-diffusion equations using a space-time spectral framework with Chebyshev collocation, enabling exponential convergence in both space and time. To confront the resulting high-dimensional systems, the authors leverage tensor-train (TT) representations and introduce a Step Truncation TT-Newton approach to regulate TT-ranks during Newton iterations, mitigating rank growth. The full-grid discretization yields a large nonlinear system, while the proposed method maintains a low-rank structure to achieve memory efficiency. Numerical demonstrations exhibit exponential convergence and substantial memory reductions relative to full-grid implementations, highlighting the method's practicality for high-fidelity, high-dimensional PDE simulations.

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.

Figures (1)

• Figure 1: 1D Space time grid with $N=4$ collocation nodes.