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

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).