An energy-conserving ultra-weak discontinuous Galerkin method for the generalized Korteweg-De Vries equation

TL;DR

This paper introduces an energy-conserving ultra-weak discontinuous Galerkin method for the generalized KdV equation, achieving optimal error estimates and superior long-time simulation performance.

Contribution

The paper develops a novel energy-conserving DG method for the generalized KdV equation with proven optimal error bounds and improved long-term stability.

Findings

Achieves optimal a priori error estimate of order k+1.

Numerical results confirm optimal convergence for equations with convection.

Demonstrates superior long-time simulation performance over existing methods.

Abstract

We propose an energy-conserving ultra-weak discontinuous Galerkin (DG) method for the generalized Korteweg-De Vries(KdV) equation in one dimension. Optimal a priori error estimate of order $k + 1$ is obtained for the semi-discrete scheme for the KdV equation without convection term on general nonuniform meshes when polynomials of degree $k\ge 2$ is used. We also numerically observed optimal convergence of the method for the KdV equation with linear or nonlinear convection terms. It is numerically observed for the new method to have a superior performance for long-time simulations over existing DG methods.