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In this paper we review the existing and develop new local discontinuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space dimensions. We review local discontinuous Galerkin methods for convection diffusion equations involving second derivatives and for KdV type equations(More)
Based on a novel numerical flux involving jumps of even order derivatives of the numerical solution, a direct discontinuous Galerkin (DDG) method for diffusion problems was introduced in [H. Liu and J. Yan, SIAM J. Numer. Anal. 47(1) (2009), 475-698]. In this work, we show that higher order (k≥4) derivatives in the numerical flux can be avoided if some(More)
In this paper we propose a new local discontinuous Galerkin method to directly solve Ham-ilton–Jacobi equations. The scheme is a natural extension of the monotone scheme. For the linear case with constant coefficients, the method is equivalent to the discontinuous Galer-kin method for conservation laws. Thus, stability and error analysis are obtained under(More)
In this paper we develop a local discontinuous Galerkin method for solving KdV type equations containing third derivative terms in one and two space dimensions. The method is based on the framework of the discontinuous Galerkin method for conservation laws and the local discontinuous Galerkin method for viscous equations containing second derivatives,(More)
A new discontinuous Galerkin finite element method for solving diffusion problems is introduced. Unlike the traditional LDG method, the scheme, called the direct discontinuous Galerkin (DDG) method, is based on the direct weak formulation for solutions of parabolic equations in each computational cell, and let cells communicate via the numerical flux ûx(More)
We develop local discontinuous Galerkin (DG) methods for solving nonlinear dispersive partial differential equations that have compactly supported traveling waves solutions, the so-called ‘‘compactons’’. The schemes we present extend the previous works of Yan and Shu on approximating solutions for linear dispersive equations and for certain KdV-type(More)
A local discontinuous Galerkin method for solving Korteweg-de Vries (KdV) type equations with non-homogeneous boundary effect is developed. We provide a criterion for imposing appropriate boundary conditions for general KdV type equations. The discussion is then focused on the KdV equation posed on the negative half plane, which arises in the modelling of(More)
We propose a discontinuous Galerkin finite element method for convection diffusion equations that involves a new methodology handling the diffusion term. Test function derivative numerical flux term is introduced in the scheme formulation to balance the solution derivative numerical flux term. The scheme has a nonsymmetric structure. For general nonlinear(More)