Symplectic QTT-FEM solution of the one-dimensional acoustic wave equation in the time domain

TL;DR

This paper introduces a novel symplectic QTT-FEM approach for solving the one-dimensional acoustic wave equation in the time domain, emphasizing energy conservation and exponential convergence.

Contribution

It develops a new QTT-FEM method combining low-rank spatial approximation with symplectic time integration for acoustic waves.

Findings

The method conserves energy effectively.

Exponential convergence observed in numerical experiments.

Applicable to time-domain acoustic wave problems.

Abstract

Structured Finite Element Methods (FEMs) based on low-rank approximation in the form of the so-called Quantized Tensor Train (QTT) decomposition (QTT-FEM) have been proposed and extensively studied in the case of elliptic equations. In this work, we design a QTT-FE method for time-domain acoustic wave equations, combining stable low-rank approximation in space with a suitable conservative discretization in time. For the acoustic wave equation with a homogeneous source term in a single space dimension as a model problem, we consider its reformulation as a first-order system in time. In space, we employ a low-rank QTT-FEM discretization based on continuous piecewise linear finite elements corresponding to uniformly refined nested meshes. Time integration is performed using symplectic high-order Gauss-Legendre Runge-Kutta methods. In our numerical experiments, we investigate the energy conservation and exponential convergence of the proposed method.