Optimally convergent HDG method for third-order Korteweg-de Vries type equations

TL;DR

This paper introduces a new hybridizable discontinuous Galerkin method for third-order Korteweg-de Vries equations, demonstrating stability and optimal convergence rates through theoretical analysis and numerical experiments.

Contribution

The paper develops a novel HDG method tailored for third-order KdV equations with proven stability and optimal error convergence, including for nonlinear cases.

Findings

Stable semi-discrete scheme with proper stabilization

Optimal convergence rates for linearized equations

Numerical experiments confirm theoretical results

Abstract

We develop and analyze a new hybridizable discontinuous Galerkin (HDG) method for solving third-order Korteweg-de Vries type equations. The approximate solutions are defined by a discrete version of a characterization of the exact solution in terms of the solutions to local problems on each element which are patched together through transmission conditions on element interfaces. We prove that the semi-discrete scheme is stable with proper choices of stabilization function in the numerical traces. For the linearized equation, we carry out error analysis and show that the approximations to the exact solution and its derivatives have optimal convergence rates. In numerical experiments, we use an implicit scheme for time discretization and the Newton-Raphson method for solving systems of nonlinear equations, and observe optimal convergence rates for both the linear and the nonlinear third-order equations.