Optimally convergent hybridizable discontinuous Galerkin method for fifth-order Korteweg-de Vries type equations
Provides a provably optimal numerical method for a class of higher-order PDEs, addressing a gap in HDG methods for fifth-order problems.
Developed the first hybridizable discontinuous Galerkin method for fifth-order KdV equations, achieving optimal convergence rates for the solution and its derivatives, validated by numerical experiments.
We develop and analyze the first hybridizable discontinuous Galerkin (HDG) method for solving fifth-order Korteweg-de Vries (KdV) type equations. We show that the semi-discrete scheme is stable with proper choices of the stabilization functions in the numerical traces. For the linearized fifth-order equations, we prove that the approximations to the exact solution and its four spatial derivatives as well as its time derivative all have optimal convergence rates. The numerical experiments, demonstrating optimal convergence rates for both the linear and nonlinear equations, validate our theoretical findings.