Optimal error estimate of the local discontinuous Galerkin methods based on the generalized alternating numerical fluxes for nonlinear convection–diffusion equations

An oscillation free local discontinuous Galerkin method for nonlinear degenerate parabolic equations

“…In this subsection, we give the optimal error estimates of the OFLDG scheme for smooth solutions of (2.1) with periodic boundary conditions and smooth initial conditions. We follow the similar approach in [11]. Since the additional damping term is used in the OFLDG scheme to control the spurious oscillations, we need to prove that the damping term would not destroy the accuracy.…”

“…Since the additional damping term is used in the OFLDG scheme to control the spurious oscillations, we need to prove that the damping term would not destroy the accuracy. Due to the nonlinear nature of the flux function h(w), we treat it by Taylor expansion as in [11,39]. Therefore, we need a priori assumption that for sufficiently small h, there holds max…”

“…In this subsection, we consider the error estimate of the OFLDG scheme (3.6)-(3.8) with the periodic boundary condition. Actually, comparing to the LDG method in [11], the proposed OFLDG scheme has an additional damping term to control the spurious oscillation. Hence, we only need to prove that the damping term does not destroy the accuracy.…”

“…We proceed to extend this approach to the convection-diffusion problems (1.1) with possibly degenerate diffusion terms. We adopt the LDG scheme in [11], in which it considered the generalized alternating numerical fluxes, which are more general and complex in the numerical analysis. Similar to [11], we can also obtain the L 2 -boundedness and optimal error estimates, while the added damping term can be estimated separately.…”

“…We adopt the LDG scheme in [11], in which it considered the generalized alternating numerical fluxes, which are more general and complex in the numerical analysis. Similar to [11], we can also obtain the L 2 -boundedness and optimal error estimates, while the added damping term can be estimated separately. It should be noted that the optimal error estimates are based on the so-called generalized Gauss-Radau (GGR) projection technique developed in [32].…”