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

TL;DR

This work develops a high-order hybridizable discontinuous Galerkin method for the two-dimensional CH–KP equation with a nonlocal term. By introducing auxiliary variables and a tensor-product discretization on Cartesian meshes, the nonlocal operator is localized, enabling efficient element-wise solves while preserving energy stability. The semi-discrete scheme is proven to be energy-stable and converges at rate $O(h^{k+1/2})$ in space for polynomial degree $k\ge2$, with numerical experiments confirming smooth-solution accuracy and the ability to resolve peakons. The approach reduces globally coupled degrees of freedom via hybridization and demonstrates robust performance on coarse grids, with potential applicability to related nonlocal dispersive systems.

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).

Figures (6)

• Figure 1: Cartesian grid structure showing the element $K_{ij}$ with its boundary faces.
• Figure 2: Error plots for $u$ (solid lines) and $q$ (dashed lines) from the accuracy test described in Section \ref{['SEC:smooth_test']}. The abscissa shows the mesh size, while the ordinate displays the $L^2$ error. Blue curves correspond to linear shape functions, red curves to quadratic shape functions, and gray curves to cubic shape functions.
• Figure 3: Cross-sectional profiles of the peakon solution using $\Delta t = 0.01$. From left to right, each curve corresponds to time $t = 0, 0.2, 0.4, 0.6, 0.8, 1.0$. Minor oscillations appear near the peak due to the larger time step, but the overall shape and location remain well captured.
• Figure 4: Cross-sectional profiles of the peakon solution using $\Delta t = 0.001$. From left to right, each curve corresponds to time $t = 0, 0.2, 0.4, 0.6, 0.8, 1.0$. The smaller time step effectively suppresses oscillations and yields a sharper, more accurate representation of the peak.
• Figure 5: Phase error for the peakon solution. The left and right panels show the error in the cross-sectional profiles along $x=0$ and $y=0$, respectively. The HDG method with $Q_2$ elements on a $64\times64$ grid and $\Delta t=0.001$ is used.

Theorems & Definitions (11)

• Lemma 3.2: Stability lemma
• proof
• Lemma 4.1: Inverse inequalities Ciarlet1978FemXuShu2007
• Lemma 4.2: Interpolation inequality cockburn2001superconvergenceCiarlet1978Fem
• Theorem 4.3: See cockburn2001superconvergenceCiarlet1978FemXuShu2007
• Theorem 4.5: A-priori $L^{2}$ error bound
• proof
• Lemma A.1
• proof
• Lemma A.2