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

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.

Figures (8)

• Figure 1: Spectral condition numbers $\kappa_2$ of matrix $\boldsymbol{Q}^{(q)}_L \boldsymbol{H}^{(q)}_L \boldsymbol{Q}^{(q)}_L$ (solid line), with $\boldsymbol{Q}^{(q)}_L$ defined in \ref{['precond']}, and $\boldsymbol{H}^{(q)}_L$ defined in \ref{['syst_k2_generalq']} (dashed line). The number of Runge--Kutta stages is $q=5$ (left plot) and $q=8$ (right plot).
• Figure 2: Horizontal axis: number of space levels L. Vertical axis: relative errors between the continuous initial data \ref{['initial_data_trig']} and their approximations from \ref{['lin_syst_u0']} and \ref{['linear_syst_v0']} when compressed in the QTT format. The error of the initial position in $\|\cdot\|_{H^1_0(\Omega)}$ (solid-blue line with round markers) decreases as $2^{-L}$ (dashed-blue line). The error of the initial velocity in $\|\cdot\|_{L^2(\Omega)}$ (solid-red line with diamond markers) decreases as $2^{-2L}$ (dashed-red line).
• Figure 3: Horizontal axis: number of space levels $L$. Vertical axis: relative errors at the final time $t_{n_t} = T$ between the exact solution \ref{['exact_sol_u']}-\ref{['exact_sol_v']} and their discrete approximations computed using Algorithm \ref{['alg:GLRK-FEM']}, when compressed in the QTT format. The number $q$ of GLRK stages is $q=1$, i.e., the implicit midpoint method is employed. The position error in $\|\cdot\|_{H^1_0(\Omega)}$ (solid-blue line with round markers) decreases as $2^{-L}$ (dashed-blue line). The velocity error in $\|\cdot\|_{L^2(\Omega)}$ (solid-red line with diamond markers) decreases as $2^{-2L}$ (dashed-red line).
• Figure 4: $L =15$ is the number of space levels, and $q=5$ is the number of GLRK stages. The step size $\tau$ of the GLRK--FEM iterations in Algorithm \ref{['alg:GLRK-FEM_precond']} satisfies Assumption \ref{['assumpt::q']}. On the horizontal axis, there are the discrete times $t_n = n\tau \in [0,1]$, $n \in \{0,\ldots,n_t\}$ with $n_t = 2^3$. Figure \ref{['fig:energy1']} compares the discrete energy $E^{(n)}_L$ (solid line) associated with the QTT-compressed GLRK--FEM iterations detailed in Algorithm \ref{['alg:GLRK-FEM_precond']}, and the total energy $E(t_n) = \frac{\pi^2}{2}$ (dashed line) of the exact solution \ref{['exact_sol_u']}-\ref{['exact_sol_v']}. Figure \ref{['fig:energy2']} shows the relative error (solid line) between the discrete energy and the exact energy. The dashed line of this figure represents the mesh size of the finite element discretization in space.
• Figure 5: $L =6$ is the number of space levels, and $q=1$ is the number of GLRK stages. Accordingly, the number of time steps $n_t$ is $n_t = 2^{L} = 2^6$. On the horizontal axis, there are the discrete times $t_n = n\tau \in [0,1]$, $n \in \{0,\ldots,n_t\}$. The plot shows the relative error (solid line) between the discrete energy $E^{(n)}_L$ associated with the QTT-compressed GLRK--FEM iterations detailed in Algorithm \ref{['alg:GLRK-FEM']}, and the total energy $E(t_n) = \frac{\pi^2}{2}$ of the exact solution \ref{['exact_sol_u']}-\ref{['exact_sol_v']}. The dashed line of this figure represents the square of the mesh size of the finite element discretization in space.

Theorems & Definitions (28)

• Definition 4.1: Quadratic invariant HLW2006
• Remark 4.2
• Lemma 4.3
• proof
• Example 4.4
• Remark 5.1
• Definition 6.1: TT decomposition O2011
• Definition 6.7: Core
• Definition 6.8: Strong Kronecker product
• Remark 6.9