Conservative and Dissipative Local Discontinuous Galerkin Methods for Korteweg-de Vries Type Equations (Communications in Computational Physics, 2018)
In this paper, we develop the Hamiltonian conservative and $L^2$ conservativelocal discontinuous Galerkin (LDG) schemes for the Korteweg-de Vries (KdV) typeequations with the minimal stencil. For the time discretization, we adopt the semi-implicitspectral deferred correction (SDC) method to achieve the high order accuracyand efficiency. Also we compare the schemes with the dissipative LDG scheme. Stabilityof the fully discrete schemes is provided by Fourier analysis for the linearized KdVequation. Numerical examples are shown to illustrate the capability of these schemes.Compared with the dissipative LDG scheme, the numerical simulations also indicatethat the conservative LDG scheme with high order time discretization can reduce thelong time phase error validly.
Keywords: Local discontinuous Galerkin method; conservative and dissipative schemes; Korteweg-de Vries type equations; semi-implicit spectral deferred correction method.