We analyze discontinuous Galerkin methods using upwind-biased numerical fluxes for time-dependent linear conservation laws. In one dimension, optimal a priori error estimates of order $k+1$ are obtained for the semidiscrete scheme when piecewise polynomials of degree at most $k$ $(k \ge 0)$ are used. Our analysis is valid for arbitrary nonuniform regular meshes and for both periodic boundary conditions and for initial-boundary value problems. We extend the analysis to the multidimensional case on Cartesian meshes when piecewise tensor product polynomials are used, and to the fully discrete scheme with explicit Runge--Kutta time discretization. Numerical experiments are shown to demonstrate the theoretical results.

The main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with implicit-explicit (IMEX) time discretization schemes, for solving one-dimensional convection-diffusion equations with a nonlinear convection. Both Runge-Kutta and multi-step IMEX methods are considered. By the aid of the energy method, we show that the IMEX LDG schemes are unconditionally stable for the nonlinear problems, in the sense that the time-step $\dt$ is only required to be upper-bounded by a positive constant which depends on the flow velocity and the diffusion coefficient, but is independent of the mesh size $h$. We also give optimal error estimates for the IMEX LDG schemes, under the same temporal condition, if a monotone numerical flux is adopted for the convection. Numerical experiments are given to verify our main results.

In this paper, we develop and analyze a new multiscale discontinuous Galerkin (DG) method for one-dimensional stationary Schr\"{o}dinger equations with open boundary conditions \bo{which have highly oscillating solutions}. Our method uses a smaller finite element space than the WKB local DG (WKB-LDG) method while achieving the same order of accuracy with no resonance errors. We prove that the DG approximation converges optimally with respect to the mesh size $h$ in $L^2$ norm without the typical constraint \bo{that $h$ has to be smaller than the wave length}. Numerical experiments were carried out to verify the second order optimal convergence rate of the method and to demonstrate its ability to capture oscillating solutions on coarse meshes in the applications to Schr\"{o}dinger equations.

We study semi-Lagrangian discontinuous Galerkin (SLDG) and Runge-Kutta discontinuous Galerkin (RKDG) schemes for some front propagation problems in the presence of an obstacle term, modeled by a nonlinear Hamilton-Jacobi equation of the form $\min(u_t + c u_x, u - g(x))=0$, in one space dimension. New convergence results and error bounds are obtained for Lipschitz regular data. These ``low regularity" assumptions are the natural ones for the solutions of the studied equations. Numerical tests are given to illustrate the behavior of our schemes.

In this paper we consider the drift-diffusion (DD) model of one dimensional semiconductor devices, which is a system involving not only first derivative convection terms but also second derivative diffusion terms and a coupled Poisson potential equation. Optimal error estimates are obtained for both the semi-discrete and fully discrete local discontinuous Galerkin (LDG) schemes with smooth solutions. In the fully discrete scheme, we couple the implicit-explicit (IMEX) time discretization with the LDG spatial discretization, in order to allow larger time steps and to save computational cost. The main technical difficulty in the analysis is to treat the inter-element jump terms which arise from the discontinuous nature of the numerical method and the nonlinearity and coupling of the models. A simulation is also performed to validate the analysis.