Comparison of Finite Volume and Discontinuous Galerkin Methods of Higher Order for Systems of Conservation Laws in Multiple Space Dimensions

The methods most frequently used in computational fluid mechanics for solving the compressible Navier-Stokes or compressible Euler equations are finite volume schemes on structured or on unstructured grids. First order as well as higher order space discretizations of MUSCL type, including flux limiters and higher order Runge- Kutta methods for the time discretization, guarantee robust and accurate schemes. But there is an important difficulty. If one increases the order, the stencil for the space discretization increases too, and the scheme becomes very expensive. Therefore schemes with more compact stencils are necessary. Discontinuous Galerkin schemes in the sense of [3] are of this type. They are identical to finite volume schemes in the case of formal first order, and for higher order they use nonconformal ansatz functions whose restrictions to single cells are polynomials of higher order. Therefore they seem to be more efficient and it is of highest interest to compare finite volume and discontinuous Galerkin methods for real applications with respect to their efficiency. Experiences [1] with the Euler equations of gas dynamics indicate that the discontinuous Galerkin methods have some advantages. Since there are no systematic studies available in the literature, we will present in this paper some numerical experiments for hyperbolic conservation laws in multiple space dimensions to compare their efficiency for different situations. As important instances of hyperbolic conservation laws we consider the Euler equations of gas dynamics and Lundquist’s equations of ideal magneto-hydrodynamics (MHD). Furthermore we have found a new limiter which improves the results from [14]. Similar studies have been done in [4].