A Hybridized Discontinuous Galerkin Method for the Nonlinear Korteweg–de Vries Equation

In this paper we introduce a hybridized discontinuous Galerkin (HDG) method for solving nonlinear Korteweg–de Vries type equations. Similar to a standard HDG implementation, we first express the approximate variables and numerical fluxes inside each element in terms of the approximate traces of the scalar variable (u), and its first derivative (\(u_x\)). These traces are assumed to be single-valued on each face. Next, we impose the conservation of numerical fluxes via two extra sets of equations. Using these global flux conservation conditions and applying the Newton–Raphson method, we construct a system of equations that can be solely expressed in terms of the increments of approximate traces in each iteration. Afterwards, we solve these equations, and substitute the approximate traces back into local equations over each element to obtain local approximate solutions. As for the time stepping scheme, we use the backward difference formulae. The method is proved to be stable for a proper choice of stabilization parameters. Through numerical examples, we observe that for a mesh with kth order elements, the computed u, p, and q show optimal convergence at order \(k+1\) in both linear and nonlinear cases, which improves upon previously employed techniques.