Optimally convergent hybridizable discontinuous Galerkin method for fifth-order Korteweg-de Vries type equations

TL;DR

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

Contribution

It is the first to develop and analyze an HDG method for fifth-order KdV equations, ensuring stability and optimal convergence rates.

Findings

Stable semi-discrete scheme with proper stabilization functions

Optimal convergence rates for solutions and derivatives

Numerical experiments confirm theoretical results

Abstract

We develop and analyze the first hybridizable discontinuous Galerkin (HDG) method for solving fifth-order Korteweg-de Vries (KdV) type equations. We show that the semi-discrete scheme is stable with proper choices of the stabilization functions in the numerical traces. For the linearized fifth-order equations, we prove that the approximations to the exact solution and its four spatial derivatives as well as its time derivative all have optimal convergence rates. The numerical experiments, demonstrating optimal convergence rates for both the linear and nonlinear equations, validate our theoretical findings.