Stability analysis of hierarchical tensor methods for time-dependent PDEs

TL;DR

This paper investigates the stability and convergence of hierarchical tensor methods combined with explicit time-stepping schemes for high-dimensional, time-dependent PDEs, highlighting limitations and providing numerical validation.

Contribution

It develops sufficient conditions for stability and convergence of tensor solutions on fixed-rank manifolds, addressing the integration of tensor methods with explicit schemes for PDEs.

Findings

Stability conditions for tensor solutions are established.

Explicit time-stepping may be limited by dimension-dependent time-step restrictions.

Numerical examples demonstrate applicability to hyperbolic and parabolic PDEs.

Abstract

In this paper we address the question of whether it is possible to integrate time-dependent high-dimensional PDEs with hierarchical tensor methods and explicit time stepping schemes. To this end, we develop sufficient conditions for stability and convergence of tensor solutions evolving on tensor manifolds with constant rank. We also argue that the applicability of PDE solvers with explicit time-stepping may be limited by time-step restriction dependent on the dimension of the problem. Numerical applications are presented and discussed for variable coefficients linear hyperbolic and parabolic PDEs.