Spectral Element-FCT Method for the Compressible Euler Equations

Abstract

A new algorithm based on spectral element discretizations and flux-corrected transport (FCT) concepts is developed for the solution of systems of hyperbolic conservation laws governing inviscid fluid flow. The current paper presents extensions to our previous work (see [6]) on scalar hyperbolic laws. A conservative formulation is proposed based on one-and two-dimensional cell-averaging and reconstruction procedures, which employ a stagerred mesh of Gauss-Chebyshev and Gauss-Lobatto-Chebyshev discretizations. Particular emphasis is placed on the construction of robust boundary and interfacial conditions in one- and two-dimensional domains. It is demonstrated through a shock-tube problem and a two-dimensional simulation of the compressible Euler equations that the proposed algorithm leads to stable, non-oscillatory solutions of high accuracy. In particular, in our model problems shocks are represented with at most two grid points, and expansion fans are represented very accurately. The spectral element-FCT method introduces minimum dispersion and diffusion errors as well as great flexibility in the discretization, since a variable number of macro-elements or collocation points per element can be employed to accomodate both accuracy and geometric requirements.

Author to whom all correspondence should be addressed