Yingda Cheng

A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives

In this paper, we develop a new discontinuous Galerkin (DG) finite element method for solving time dependent partial differential equations (PDEs) with higher order spatial derivatives. Unlike the traditional local discontinuous Galerkin (LDG) method, the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system. Stability is ensured by a careful choice of interface numerical fluxes. The method can be designed for quite general nonlinear PDEs and we prove stability and give error estimates for a few representative classes of PDEs up to fifth order. Numerical examples show that our scheme attains the optimal (k + 1)-th order of accuracy when using piecewise k-th degree polynomials, under the condition that k + 1 is greater than or equal to the order of the equation.

Superconvergence of Discontinuous Galerkin and Local Discontinuous Galerkin Schemes for Linear Hyperbolic and Convection-Diffusion Equations in One Space Dimension

In this paper, we study the superconvergence property for the discontinuous Galerkin (DG) and the local discontinuous Galerkin (LDG) methods for solving one-dimensional time dependent linear conservation laws and convection-diffusion equations. We prove superconvergence towards a particular projection of the exact solution when the upwind flux is used for conservation laws and when the alternating flux is used for convection-diffusion equations. The order of superconvergence for both cases is proved to be

A discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations

k+\frac{3}{2}

Study of conservation and recurrence of Runge---Kutta discontinuous Galerkin schemes for Vlasov---Poisson systems

when piecewise

Superconvergence and time evolution of discontinuous Galerkin finite element solutions

P^k

Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations

polynomials with

Discontinuous Galerkin Methods for the Vlasov--Maxwell Equations

k\geq1

Energy-conserving discontinuous Galerkin methods for the Vlasov-Ampère system

are used. The proof is valid for arbitrary nonuniform regular meshes and for piecewise

A discontinuous Galerkin scheme for front propagation with obstacles

P^k

A Discontinuous Galerkin Solver for Full-Band Boltzmann-Poisson Models

polynomials with arbitrary