arrow
Volume 14, Issue 1
A Diagonalization-Based Parallel-in-Time Algorithm for Crank-Nicolson’s Discretization of the Viscoelastic Equation

Fu Li & Yingxiang Xu

East Asian J. Appl. Math., 14 (2024), pp. 47-78.

Published online: 2024-01

Export citation
  • Abstract

In this paper, we extend a diagonalization-based parallel-in-time (PinT) algorithm to the viscoelastic equation. The central difference method is used for spatial discretization, while for temporal discretization, we use the Crank-Nicolson scheme. Then an all-at-once system collecting all the solutions at each time level is formed and solved using a fixed point iteration preconditioned by an $α$-circulant matrix in parallel. Via a rigorous analysis, we find that the spectral radius of the iteration matrix is uniformly bounded by $α/(1 − α),$ independent of the model parameters (the damping coefficient $\varepsilon$ and the wave velocity $\sqrt{\gamma}$) and the discretization parameters (the time step $\tau$ and the spatial mesh size $h$). Unlike the classical wave equation with Dirichlet boundary condition where the upper bound $α/(1 − α)$ is very sharp, we find that the occurrence of the damping term $−\varepsilon∆y_t,$ as well as the large final time $T,$ leads to even faster convergence of the algorithm, especially when $α$ is not very small. We illustrate our theoretical findings with several numerical examples.

  • AMS Subject Headings

65F08, 65F10, 65M22, 65Y05

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{EAJAM-14-47, author = {Li , Fu and Xu , Yingxiang}, title = {A Diagonalization-Based Parallel-in-Time Algorithm for Crank-Nicolson’s Discretization of the Viscoelastic Equation}, journal = {East Asian Journal on Applied Mathematics}, year = {2024}, volume = {14}, number = {1}, pages = {47--78}, abstract = {

In this paper, we extend a diagonalization-based parallel-in-time (PinT) algorithm to the viscoelastic equation. The central difference method is used for spatial discretization, while for temporal discretization, we use the Crank-Nicolson scheme. Then an all-at-once system collecting all the solutions at each time level is formed and solved using a fixed point iteration preconditioned by an $α$-circulant matrix in parallel. Via a rigorous analysis, we find that the spectral radius of the iteration matrix is uniformly bounded by $α/(1 − α),$ independent of the model parameters (the damping coefficient $\varepsilon$ and the wave velocity $\sqrt{\gamma}$) and the discretization parameters (the time step $\tau$ and the spatial mesh size $h$). Unlike the classical wave equation with Dirichlet boundary condition where the upper bound $α/(1 − α)$ is very sharp, we find that the occurrence of the damping term $−\varepsilon∆y_t,$ as well as the large final time $T,$ leads to even faster convergence of the algorithm, especially when $α$ is not very small. We illustrate our theoretical findings with several numerical examples.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.2022-304.070323}, url = {http://global-sci.org/intro/article_detail/eajam/22319.html} }
TY - JOUR T1 - A Diagonalization-Based Parallel-in-Time Algorithm for Crank-Nicolson’s Discretization of the Viscoelastic Equation AU - Li , Fu AU - Xu , Yingxiang JO - East Asian Journal on Applied Mathematics VL - 1 SP - 47 EP - 78 PY - 2024 DA - 2024/01 SN - 14 DO - http://doi.org/10.4208/eajam.2022-304.070323 UR - https://global-sci.org/intro/article_detail/eajam/22319.html KW - Parallel-in-time (PinT) algorithm, Crank-Nicolson method, diagonalization, viscoelastic equation, convergence analysis. AB -

In this paper, we extend a diagonalization-based parallel-in-time (PinT) algorithm to the viscoelastic equation. The central difference method is used for spatial discretization, while for temporal discretization, we use the Crank-Nicolson scheme. Then an all-at-once system collecting all the solutions at each time level is formed and solved using a fixed point iteration preconditioned by an $α$-circulant matrix in parallel. Via a rigorous analysis, we find that the spectral radius of the iteration matrix is uniformly bounded by $α/(1 − α),$ independent of the model parameters (the damping coefficient $\varepsilon$ and the wave velocity $\sqrt{\gamma}$) and the discretization parameters (the time step $\tau$ and the spatial mesh size $h$). Unlike the classical wave equation with Dirichlet boundary condition where the upper bound $α/(1 − α)$ is very sharp, we find that the occurrence of the damping term $−\varepsilon∆y_t,$ as well as the large final time $T,$ leads to even faster convergence of the algorithm, especially when $α$ is not very small. We illustrate our theoretical findings with several numerical examples.

Li , Fu and Xu , Yingxiang. (2024). A Diagonalization-Based Parallel-in-Time Algorithm for Crank-Nicolson’s Discretization of the Viscoelastic Equation. East Asian Journal on Applied Mathematics. 14 (1). 47-78. doi:10.4208/eajam.2022-304.070323
Copy to clipboard
The citation has been copied to your clipboard