In this paper, we develop the Hamiltonian conservative and L2 conservative local discontinuous Galerkin (LDG) schemes for the Korteweg-de Vries (KdV) type equations with the minimal stencil. For the time discretization, we adopt the semi- implicit spectral deferred correction (SDC) method to achieve the high order accuracy and efficiency. Also we compare the schemes with the dissipative LDG scheme. Stabil- ity of the fully discrete schemes is provided by Fourier analysis for the linearized KdV equation. Numerical examples are shown to illustrate the capability of these schemes. Compared with the dissipative LDG scheme, the numerical simulations also indicate that the conservative LDG scheme with high order time discretization can reduce the long time phase error validly.

H(div) conforming and discontinuous Galerkin (DG) methods are designed for incompressible Euler's equation in two and three dimension. Error estimates are proved for both the semi-discrete method and fully-discrete method using backward Euler time stepping. Numerical examples exhibiting the performance of the methods are given.

In this paper, we generalize the point integral method to solve the general elliptic PDEs with variable coefficients and corresponding eigenvalue problems with Neumann, Robin and Dirichlet boundary conditions on point cloud. The main idea is using integral equations to approximate the original PDEs. The integral equations are easy to discretize on the point cloud. The truncation error of the integral approximation is analyzed. Numerical examples are presented to demonstrate that PIM is an effective method to solve the elliptic PDEs with smooth coefficients on point cloud.

In this paper, we study the superconvergence of the error for the discontinuous Galerkin (DG) finite element method for linear conservation laws when upwind fluxes are used. We prove that if we apply piecewise $k$-th degree polynomials, the error between the DG solution and the exact solution is ($k+2$)-th order superconvergent at the downwind-biased Radau points with suitable initial discretization. Moreover, we also prove the DG solution is ($k+2$)-th order superconvergent both for the cell averages and for the error to a particular projection of the exact solution. The superconvergence result in this paper leads to a new {\em a posteriori} error estimate. Our analysis is valid for arbitrary regular meshes and for $\mathcal{P}^k$ polynomials with arbitrary $k\geq1$, and for both periodic boundary conditions and for initial-boundary value problems. We perform numerical experiments to demonstrate that the superconvergence rate proved in this paper is optimal.

In this paper, we develop the high order weighted essentially non-oscillatory (WENO) schemes for solving the Degasperis-Procesi (DP) equation, including finite volume (FV) and finite difference (FD) methods. The DP equation contains nonlinear high order derivatives, and possibly discontinuous or sharp transition solutions. The finite volume method is de- signed based on the total variation bounded property of the DP equation. And the finite difference method is constructed based on the L2 stability of the DP equation. Due to the adoption of the WENO reconstruction, both schemes are arbitrary high order accuracy and shock capturing. The numerical simulation results for different types of solutions of the nonlinear Degasperis-Procesi equation are provided to illustrate the accuracy and capability of the methods.