Conservative and Dissipative Local Discontinuous Galerkin Methods for Korteweg-de Vries Type Equations

Abstract

In this paper, we develop the Hamiltonian conservative and L 2 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 semiimplicit spectral deferred correction (SDC) method to achieve the high order accuracy and efficiency. Also we compare the schemes with the dissipative LDG scheme. Stability 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.

Discontinuous Galerkin methods for short pulse type equations via hodograph transformations

“…In the numerical test Example 4.2, it shows that the dissipative scheme with parameters (2.38) can achieve the optimal convergence rate for both u h and ρ h no matter k is odd or even. However, the order of accuracy for the H 0 conserved DG scheme is k-th for odd k, (k + 1)-th for even k. The choices of these parameters are not unique, but the above numerical fluxes in the dissipative scheme can minimize the stencil as in [42].…”

“…In what follows, we prepare to give the a priori error estimate for the H 1 conserved DG scheme. The standard L 2 projection of a function ζ with k + 1 continuous derivatives into 42) and the special projections P ± into V k h satisfy, for each I j…”

A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations

“…All of the previously existing conservative numerical methods such as [50,63,64,105,111] are constructed using ad hoc techniques tailored specifically to both the equation and the numerical method. Typically, the ideas used to construct one such scheme cannot immediately be adapted to another equation or type of discretization.…”

Discontinuous Galerkin Methods for the Ostrovsky–Vakhnenko Equation

“…As the conservative methods for KdV equation [3,14,40], Zakharov system [34], Schrödinger-KdV system [35], short pulse equation [41], etc., various conservative numerical schemes are proposed to "preserve structure". Usually, the conservative schemes can help reduce the phase error along the long time evolution.…”