Abstract
In this paper, we study the discontinuous Galerkin (DG) method with a class of generalized numerical fluxes for one-dimensional scalar nonlinear conservation laws. The generalized local Lax--Friedrichs (GLLF) fluxes with two weights, which may not be monotone, are proposed and analyzed. Under a condition for the weights, we first show the monotonicity for the flux and thus the $L^2$ stability of the scheme. Then, by constructing and analyzing a special piecewise global projection which commutes with the time derivative operator, we are able to show optimal error estimates for the DG scheme with GLLF fluxes. The result is sharp for monotone numerical fluxes, for which only suboptimal estimates can be proved in previous work. Moreover, optimal error estimates are still valid for fluxes that are not monotone, allowing us to choose some suitable weights to achieve less numerical dissipation and thus to better resolve shocks. Numerical experiments are provided to show the sharpness of theoretical results.
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