Abstract—This paper deals with a high-order accurate Nodal Discontinuous Galerkin (DG) method for the numerical solution of the inviscid Burgers equation, which is a simplest case of nonlinear, hyperbolic partial differential equation. This method combines mainly two key ideas which are based on the finite volume and finite element methods. The physics of wave propagation being accounted for by means of Riemann problems and accuracy is obtained by means of high-order polynomial approximations within elements. In Nodal DG method a finite element space discretization is obtained by element wise discontinuous approximations. Whereas low-storage, high order accurate, explicit Runge-Kutta (LSERK) method is used for temporal discretization. The resulting RKDG methods are stable, high-order accurate and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. Exponential filter is used to remove spurious oscillations near the shock waves. The   and   errorsin the solution show that the scheme is accurate and effective. Hence, the method is well suited to achieve high order accurate solution for the hyperbolic partial differential equations. Index Terms—Nodal Discontinuous galerkin method, burgers equation, exponential filter, hyperbolic PDE. I. INTRODUCTION The Discontinuous Galerkin Method (DGM) was first introduced by Reed and Hill [1] as a technique to solve neutron transport problems. In a series of papers by Cockburn, Shu et al. [2-5], the RKDG method has been developed for solving nonlinear hyperbolic conservation laws and related equations, in which DG is used for spatial discretization with flux values at cell edges computed by either Riemann solversor monotone flux functions, the total variation bounded (TVB) limiter [6-7] is employedto eliminate spurious oscillations and the total variation diminishing (TVD) Runge-Kutta(RK) method is used for the temporal discretization to ensure the stability of the numericalapproach while simplifying the implementation. The Discontinuous Galerkin Method (DGM) has recently become more popular for the solution of systems of conservation laws to arbitrary order of accuracy [8], [9]. An intelligent combination of the finite element and finite volume method, utilizing a space of basis and test function that mimics thefinite element method but satisfying the Manuscript received May 18, 2012; revised June 12, 2012 The authors are with the Northwestern Polytechnical University, Xi’ an, China (Tel.: +8618629067124; e-mail: fareedmuet@yahoo.com). equation in a sense closer to the finite volume method, appears to offer many of the desired properties. This combination is exactly what leads to Discontinuous Galerkin Finite Element Method (DG-FEM) [10]. The physics of wave propagation is, however, accounted for by solving the Riemann problems that arise from the discontinuous representation of the solution at element interfaces [11], [12], [13]. High order accurate Low-Storage Explicit Runge-Kutta (LSERK) method is used for temporaldiscretization.The resultingRKDG methods are stable, high-order accurate and highly parallelizable schemesthat can easily handle complicated geometries and boundary conditions [14]. Many of the equations of mechanics are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. II. NODAL DISCONTINUOUS GALERKIN FORMULATION A. Spatial Discretization The conservation law is discretized in space by using Discontinuous Galerkin approach. Here we consider problem posed on the physical domain  with boundary  and assume that this domain is well approximated by the computational domain   . This is space filling triangulation composed of a collection of k geometry–conforming non-overlapping elements  . The model equation is             where the flux is given as      .It is subject to the initial conditions and boundary conditions of the form:                        We approximate  by K non-overlapping elements,            as illustrated in Fig. 1. On each of these elements we express the local solution as a polynomial of order                                   A High Order Accurate Nodal Discontinuous Galerkin Method (DGM) for Numerical Solution of Hyperbolic Equation International Journal of Applied Physics and Mathematics, Vol. 2, No. 5, September 2012 362 Fareed Ahmed, Faheem Ahmed,Yonghang Guo, and Yong Yang