Superconvergence of the Local Discontinuous Galerkin Method with Generalized Numerical Fluxes for One-Dimensional Nonlinear Time-Dependent Fourth-Order Equations

In this paper, we concentrate on the superconvergence of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear time-dependent fourth-order equations. The numerical flux for the nonlinear convection term is chosen as the generalized local Lax–Friedrichs flux, and the generalized alternating fluxes are employed for the fourth- and second-order terms, which are beneficial for long time simulations with a slower error growth due to the adjustable numerical viscosities. For nonlinear fourth-order equations with periodic boundary conditions, by using generalized Gauss–Radau projections, a modified projection and correction functions, we show a superconvergent bound for the interpolation errors. Then, by designing the numerical initial condition as an interpolation function of the third-order derivative, we derive supercloseness and thus superconvergence results, no matter whether the wind direction is fixed or not. Specifically, for polynomials of degree k, we obtain (2k+1)th order superconvergence for the numerical flux and cell averages, (k+2)th order superconvergence at generalized Radau points, and (k+1)th order for the error derivative at generalized Radau points, followed by a supercloseness result of order k+2 between the generalized Gauss–Radau projections and the numerical solutions. The superconvergence results are extended to the case with mixed boundary conditions when the wind direction is fixed. A series of numerical examples, including various boundary conditions and nonlinear terms, together with long time simulations, are provided to validate the theoretical results and demonstrate the effectiveness of the method.