Optimally convergent HDG method for third-order Korteweg-de Vries type equations
This work provides a provably optimal numerical method for a class of nonlinear PDEs, which is incremental as it extends HDG methods to third-order equations.
The authors develop a new hybridizable discontinuous Galerkin method for third-order Korteweg-de Vries type equations, proving optimal convergence rates for the semi-discrete scheme and demonstrating optimal rates in numerical experiments for both linear and nonlinear cases.
We develop and analyze a new hybridizable discontinuous Galerkin (HDG) method for solving third-order Korteweg-de Vries type equations. The approximate solutions are defined by a discrete version of a characterization of the exact solution in terms of the solutions to local problems on each element which are patched together through transmission conditions on element interfaces. We prove that the semi-discrete scheme is stable with proper choices of stabilization function in the numerical traces. For the linearized equation, we carry out error analysis and show that the approximations to the exact solution and its derivatives have optimal convergence rates. In numerical experiments, we use an implicit scheme for time discretization and the Newton-Raphson method for solving systems of nonlinear equations, and observe optimal convergence rates for both the linear and the nonlinear third-order equations.