A Class of Efficient Hamiltonian Conservative Spectral Methods for Korteweg-de Vries Equations

In this paper, we present and introduce two efficient Hamiltonian conservative fully discrete numerical schemes for Korteweg-de Vries equations. The new numerical schemes are constructed by using time-stepping spectral Petrov-Galerkin (SPG) or Gauss collocation (SGC) methods for the temporal discretization coupled with the p-version/spectral local discontinuous Galerkin (LDG) methods for the space discretization. We prove that the fully discrete SPG-LDG scheme preserves both the momentum and the Hamilton energy exactly for generalized KdV equations. While the fully discrete SGC-LDG formulation preserves the momentum and the Hamilton energy exactly for linearized KdV equations. As for nonlinear KdV equations, the SGC-LDG scheme preserves the momentum exactly and is Hamiltonian conserving up to some spectral accuracy. Furthermore, we show that the fully-discrete SGC-LDG and SPG-LDG methods converge exponentially with respect to the polynomial degree in both space and time for linear KdV equations. The numerical experiments are provided to demonstrate that the proposed numerical methods preserve the momentum, \(L^2\) energy and Hamilton energy and maintain the shape of the solution phase efficiently over long time period.