Optimal Error Estimates of the Local Discontinuous Galerkin Method with Generalized Numerical Fluxes for One-Dimensional KdV Type Equations (Journal of Computational Mathematics, 2024)
In this paper, we investigate the local discontinuous Galerkin method with generalizednumerical fluxes for one-dimensional nonlinear Korteweg-de Vries type equations. Thenumerical flux for the nonlinear convection term is chosen as the generalized Lax-Friedrichsflux, and the generalized alternating flux and upwind-biased flux are used for the dispersionterm. The generalized Lax-Friedrichs flux with anti-dissipation property will compensatethe numerical dissipation of the dispersion term, resulting in a nearly energy conservativescheme that is useful in resolving waves and is beneficial for long time simulations. Todeal with the nonlinearity and different numerical flux weights, a suitable numerical initialcondition is constructed, for which a modified global projection is designed. By establishingrelationships between the prime variable and auxiliary variables in combination with sharpbounds for jump terms, optimal error estimates are obtained. Numerical experiments areshown to confirm the validity of theoretical results.