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In this paper, we present optimal error estimates of the local discontinuousGalerkin method with generalized numerical fluxes for one-dimensional nonlinearconvection-diffusion systems. The upwind-biased flux with adjustable numericalviscosity for the convective term is chosen based on the local characteristicdecomposition, which is helpful in resolving discontinuities of degenerateparabolic equations without enforcing any limiting procedure. For the diffusiveterm, a pair of generalized alternating fluxes are considered. By constructingand analyzing generalized Gauss-Radau projections with respect to differentconvective or diffusive terms, we derive optimal error estimates for nonlinearconvection-diffusion systems with the symmetrizable flux Jacobian and fullynonlinear diffusive problems. Numerical experiments including long timesimulations, different boundary conditions and degenerate equations withdiscontinuous initial data are provided to demonstrate the sharpness oftheoretical results.

Analysis of the local discontinuous Galerkin method with generalized fluxes for 1D nonlinear convection-diffusion systems