Superconvergence of the Local Discontinuous Galerkin Method With Generalized Numerical Fluxes for One‐Dimensional KdV‐Type Equations

ABSTRACT

In this paper, we study superconvergence properties of the local discontinuous Galerkin (LDG) method with generalized numerical fluxes for one‐dimensional Korteweg–de Vries (KdV)‐type equations. Compared with traditional upwind and alternating fluxes, the LDG solution using generalized numerical fluxes exhibits a slower error growth for long‐time simulations. The main technique involves the construction of a new modified projection and the estimation of projection errors. When the direction of the convection wind is fixed, by the properties of correction functions and appropriate numerical initial conditions, superconvergence results are established, including the th order for cell averages as well as numerical fluxes, and the th (th) order for the LDG error (its derivative) at generalized Radau points. Numerical experiments for the nonlinear flux with the generalized Lax–Friedrichs flux and mixed boundary conditions, as well as other boundaries, are carried out. All theoretical findings are confirmed by numerical experiments.