Symplectic QTT-FEM solution of the one-dimensional acoustic wave equation in the time domain
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917840471195648 |
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| author | Fraschini, Sara Kazeev, Vladimir Perugia, Ilaria |
| author_facet | Fraschini, Sara Kazeev, Vladimir Perugia, Ilaria |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_11321 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symplectic QTT-FEM solution of the one-dimensional acoustic wave equation in the time domain Fraschini, Sara Kazeev, Vladimir Perugia, Ilaria Numerical Analysis 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. |
| title | Symplectic QTT-FEM solution of the one-dimensional acoustic wave equation in the time domain |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2411.11321 |