Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations

In this paper, we provide $L^2$ error estimates for the semi-discrete local discontinuous Galerkin methods for nonlinear convection-diffusion equations and KdV equations with smooth solutions. The main technical difficulty is the control of the inter-element jump terms which arise because of the nonlinearity of the PDEs and the discontinuous nature of the numerical method