Discontinuous Galerkin Method For First Order Hyperbolic Problems

First order hyperbolic problems have a wide range of sources. In practical applications, such as water conservancy, meteorology, aerospace, molecular and any other fields of fluid mechanics problems can be attributed to solving a first-order hyperbolic equation or first-order hyperbolic equations.Traditionally, the numerical method for solving first order hyperbolic problem is finite difference method. Finite difference method is simple and the structure can be implemented easily, but it is hard to deal with irregular area or complex boundary conditions. The high-precision differential format of it is hard to construct and the results need a strong smooth in order to do error analysis. Finite element method is precisely to overcome these shortcomings, but the traditional finite element method for solving first order hyperbolic problems is not very effective. People therefore introduced some kinds of new forms of finite element method, discontinuous Galerkin method is one kind of effective numerical methods.The essence of numerical methods for partial differential equations is the finite dimensional space approximating the infinite dimensional space. Thus, the problem which is belonging to the infinite dimensional space is discretized into a finite dimensional approximation problem. Discontinuous Galerkin method is a kind of finite element methods which uses completely interrupted polynomial space as test space. In this paper, we first discuss the typical first-order hyperbolic equation and construct a discretized scheme by discontinuous Galerkin method, then, we construct an improvement scheme by using penalty factor to enhance the stability of the scheme. Further, the theoretical analysis of the discontinuous Galerkin method scheme is given. We derive a prior error estimate and establish a posteriori error estimate by use a comparison function. Analyzing and comparing the numerical results which solved by finite difference method and the discontinuous Galerkin method, we verified the effective of the discontinuous Galerkin method.