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.

翻译：谱方法为偏微分方程提供高精度数值解，在谱节点数量上呈现指数收敛性。传统上，在处理时间相关的非线性问题时，关注点集中于时间离散化的低阶有限差分格式和空间变量的谱元格式。然而，我们近期的研究进展实现了谱方法在空间与时间变量上的同时应用，从而在两个维度上均保持了谱收敛特性。通过运用张量链技术，我们的方法克服了时空方法固有的维度灾难问题。本文将此方法推广至非线性瞬态对流扩散方程。我们的离散化格式呈现出低秩结构，便于转换为张量链格式。然而，在处理非线性问题所需的牛顿迭代过程中控制张量链秩面临挑战，为此我们设计了"步进截断张量链-牛顿"方法。通过多个基准算例，我们验证了该方法具有指数收敛性。尤为重要的是，相较于全网格格式，本方案显著降低了内存需求。