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

TL;DR

This paper provides optimal error estimates for a local discontinuous Galerkin method with generalized fluxes applied to 1D nonlinear convection-diffusion systems, improving accuracy and handling discontinuities without limiting procedures.

Contribution

It introduces a novel analysis of the LDG method with generalized fluxes, including new error estimates and flux choices for nonlinear convection-diffusion equations.

Findings

Optimal error estimates derived for the method.

Numerical experiments confirm theoretical sharpness.

Effective handling of discontinuities without limiting procedures.

Abstract

In this paper, we present optimal error estimates of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear convection-diffusion systems. The upwind-biased flux with adjustable numerical viscosity for the convective term is chosen based on the local characteristic decomposition, which is helpful in resolving discontinuities of degenerate parabolic equations without enforcing any limiting procedure. For the diffusive term, a pair of generalized alternating fluxes are considered. By constructing and analyzing generalized Gauss-Radau projections with respect to different convective or diffusive terms, we derive optimal error estimates for nonlinear convection-diffusion systems with the symmetrizable flux Jacobian and fully nonlinear diffusive problems. Numerical experiments including long time simulations, different boundary conditions and degenerate equations with discontinuous initial data are provided to demonstrate the sharpness of theoretical results.