Fast Solver for the Local Discontinuous Galerkin Discretization of the KdV Type Equations (Communications in Computational Physics, 2020)
In this paper, we will develop a fast iterative solver for the system of linearequations arising from the local discontinuous Galerkin (LDG) spatial discretizationand 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 thepartial 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 numericallyshow the optimal or sub-optimal complexity of the MG solver and a two-levellocal mode analysis is used to analyze the convergence behavior of the MG method.Numerical results for one-dimensional, two-dimensional and three-dimensional casesare given to illustrate the efficiency and capability of the LDG method coupled withthe multigrid method for solving the KdV type equations.