A Hybridizable Discontinuous Galerkin Method for the non--local Camassa--Holm--Kadomtsev--Petviashvili equation

TL;DR

This paper introduces a hybridizable discontinuous Galerkin method for a complex two-dimensional non-local PDE, achieving stability and convergence, and effectively capturing both smooth and peaked wave solutions.

Contribution

The paper presents a novel HDG method tailored for the non-local 2D Camassa-Holm--Kadomtsev--Petviashvili equation, including localization of non-local operators and stability analysis.

Findings

Proves energy stability of the semi-discrete scheme

Achieves h^{k+1/2} convergence rate in space

Successfully captures smooth and peaked solitary waves

Abstract

This paper develops a hybridizable discontinuous Galerkin method for the two-dimensional Camassa--Holm--Kadomtsev--Petviashvili equation. The method employs Cartesian meshes with tensor-product polynomial spaces, enabling separate treatment of \(x\) and \(y\) derivatives. The non-local operator \(\partial_{x}^{-1}u_{y}\) is localized through an auxiliary variable \(v\) satisfying \(v_x = u_y\), allowing efficient element-by-element computations. We prove energy stability of the semi-discrete scheme and derive \(\mathcal{O}(h^{k+1/2})\) convergence in space. Numerical experiments validate the theoretical results and demonstrate the method's capability to accurately resolve smooth solutions and peaked solitary waves (peakons).