Superconvergence of the Local Discontinuous Galerkin Method for One Dimensional Nonlinear Convection-Diffusion Equations

Abstract

In this paper, we study superconvergence properties of the local discontinuous Galerkin (LDG) methods for solving nonlinear convection-diffusion equations in one space dimension. The main technicality is an elaborate estimate to terms involving projection errors. By introducing a new projection and constructing some correction functions, we prove the $$(2k+1)$$ ( 2 k + 1 ) th order superconvergence for the cell averages and the numerical flux in the discrete $$L^2$$ L 2 norm with polynomials of degree $$k\\ge 1$$ k ≥ 1 , no matter whether the flow direction $$f (u)$$ f ′ ( u ) changes or not. Superconvergence of order $$k +2$$ k + 2 ( $$k +1$$ k + 1 ) is obtained for the LDG error (its derivative) at interior right (left) Radau points, and the convergence order for the error derivative at Radau points can be improved to $$k+2$$ k + 2 when the direction of the flow doesn’t change. Finally, a supercloseness result of order $$k+2$$ k + 2 towards a special Gauss–Radau projection of the exact solution is shown. The superconvergence analysis can be extended to the generalized numerical fluxes and the mixed boundary conditions. All theoretical findings are confirmed by numerical experiments.