Devising discontinuous Galerkin methods for non-linear hyperbolic conservation laws

In this paper, we give a simple introduction to the devising of discontinuous Galerkin (DG) methods for nonlinear hyperbolic conservation laws. These methods have recently made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. The DG methods, which are extensions of finite volume methods, incorporate into a finite element framework the notions of approximate Riemann solvers, numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume methods for nonlinear hyperbolic conservation laws. We start by stressing the fact that nonlinear hyperbolic conservation laws are usually obtained from well-posed problems by neglecting terms modeling nondominant features of the model which, nevertheless, are essential in crucial, small parts of the domain; as a consequence, the resulting problem becomes ill-posed. The main difficulty in devising numerical schemes for these conservation laws is thus how to re-introduce the neglected physical information in order to approximate the physically relevant solution, usually called the entropy solution. For the classical case of the entropy solution of the nonlinear hyperbolic scalar conservation law, we show how to carry out this process for two prototypical DG methods. The first DG method is the so-called shock-capturing DG method, which does not use slope limiters and is implicit; the second is the Runge–Kutta DG method, which is an explicit method that does not employ a shock-capturing term but uses a slope limiter instead. We then focus on the Runge–Kutta DG methods and show how to obtain a key stability property which holds independently of the accuracy of the scheme and of the nonlinearity of the conservation law; we also show some computational results.