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

In this paper, we study the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear time-dependent fourth-order equations. The L2 stability and optimal error estimates of order k+1 are obtained with piecewise polynomials of degree k≥1. The numerical flux of the nonlinear convection term is the generalized local Lax–Friedrichs flux, and the generalized alternating fluxes are adopted for the fourth- and second-order terms. The adjustable numerical viscosity of these fluxes is beneficial for long time simulations with a slower error growth. A proper numerical initial condition is designed based on a modified projection of the third-order derivative. By using the generalized Gauss–Radau projections, together with sharp bounds of nonlinear and jump terms, optimal error estimates are derived. The results are extended to equations with an additional dispersion term and mixed boundary conditions. Examples including Dirichlet as well as generalized Dirichlet boundary conditions, singularly perturbed problems and two-dimensional problems are also numerically investigated, indicating that the theoretical results hold for more general cases.