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 , 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.

We consider the development and analysis of local discontinuous Galerkin methods forfractional diffusion problems in one space dimension, characterized by having fractionalderivatives, parameterized by β ∈[1, 2]. After demonstrating that aclassic approach fails to deliver optimal order of convergence, we introduce a modifiedlocal numerical flux which exhibits optimal order of convergence 𝒪(hk + 1) uniformly across the continuous range between pureadvection (β = 1) and pure diffusion (β = 2). In the twoclassic limits, known schemes are recovered. We discuss stability and present an erroranalysis for the space semi-discretized scheme, which is supported through a fewexamples.