Volume 17, Issue 2
Fast Solver for the Local Discontinuous Galerkin Discretization of the KdV Type Equations

Ruihan Guo & Yan Xu

Commun. Comput. Phys., 17 (2015), pp. 424-457.

Published online: 2018-04

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  • Abstract

In this paper, we will develop a fast iterative solver for the system of linear equations arising from the local discontinuous Galerkin (LDG) spatial discretization and additive Runge-Kutta (ARK) time marching method for the KdV type equations. Being implicit in time, the severe time step ($∆t$=$\mathcal{O}(∆x^k)$, with the $k$-th order of the partial differential equations (PDEs)) restriction for explicit methods will be removed. The equations at the implicit time level are linear and we demonstrate an efficient, practical multigrid (MG) method for solving the equations. In particular, we numerically show the optimal or sub-optimal complexity of the MG solver and a two-level local mode analysis is used to analyze the convergence behavior of the MG method. Numerical results for one-dimensional, two-dimensional and three-dimensional cases are given to illustrate the efficiency and capability of the LDG method coupled with the multigrid method for solving the KdV type equations.

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@Article{CiCP-17-424, author = {}, title = {Fast Solver for the Local Discontinuous Galerkin Discretization of the KdV Type Equations}, journal = {Communications in Computational Physics}, year = {2018}, volume = {17}, number = {2}, pages = {424--457}, abstract = {

In this paper, we will develop a fast iterative solver for the system of linear equations arising from the local discontinuous Galerkin (LDG) spatial discretization and additive Runge-Kutta (ARK) time marching method for the KdV type equations. Being implicit in time, the severe time step ($∆t$=$\mathcal{O}(∆x^k)$, with the $k$-th order of the partial differential equations (PDEs)) restriction for explicit methods will be removed. The equations at the implicit time level are linear and we demonstrate an efficient, practical multigrid (MG) method for solving the equations. In particular, we numerically show the optimal or sub-optimal complexity of the MG solver and a two-level local mode analysis is used to analyze the convergence behavior of the MG method. Numerical results for one-dimensional, two-dimensional and three-dimensional cases are given to illustrate the efficiency and capability of the LDG method coupled with the multigrid method for solving the KdV type equations.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.210114.080814a}, url = {http://global-sci.org/intro/article_detail/cicp/10964.html} }
TY - JOUR T1 - Fast Solver for the Local Discontinuous Galerkin Discretization of the KdV Type Equations JO - Communications in Computational Physics VL - 2 SP - 424 EP - 457 PY - 2018 DA - 2018/04 SN - 17 DO - http://doi.org/10.4208/cicp.210114.080814a UR - https://global-sci.org/intro/article_detail/cicp/10964.html KW - AB -

In this paper, we will develop a fast iterative solver for the system of linear equations arising from the local discontinuous Galerkin (LDG) spatial discretization and additive Runge-Kutta (ARK) time marching method for the KdV type equations. Being implicit in time, the severe time step ($∆t$=$\mathcal{O}(∆x^k)$, with the $k$-th order of the partial differential equations (PDEs)) restriction for explicit methods will be removed. The equations at the implicit time level are linear and we demonstrate an efficient, practical multigrid (MG) method for solving the equations. In particular, we numerically show the optimal or sub-optimal complexity of the MG solver and a two-level local mode analysis is used to analyze the convergence behavior of the MG method. Numerical results for one-dimensional, two-dimensional and three-dimensional cases are given to illustrate the efficiency and capability of the LDG method coupled with the multigrid method for solving the KdV type equations.

Ruihan Guo & Yan Xu. (2020). Fast Solver for the Local Discontinuous Galerkin Discretization of the KdV Type Equations. Communications in Computational Physics. 17 (2). 424-457. doi:10.4208/cicp.210114.080814a
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