Artificial dissipation models for the Euler equations

Abstract

Various artificial dissipation models that are used with central difference algorithms for the Euler equations are analyzed for their effect on accuracy, stability, and convergence rates. In particular, linear and nonlinear models are investigated using an implicit approximate factorization code (ARC2D) for transonic airfoils. Fully implicit application of the dissipation models is shown to improve robustness and convergence rates. The treat- ment of dissipation models at boundaries will be examined. It will be shown that accurate, error free solutions with sharp shocks can be obtained using a central difference algorithm coupled with an appropriate nonlinear artificial dissipation model. I. Introduction T HE solution of the Euler equations using numerical tech- niques requires the use of either a differencing method with inherent dissipation or the addition of dissipation terms to a nondissipative scheme. This is because the Euler equa- tions do not provide any natural dissipation mechanism (such as viscosity in the Navier-Stokes equations) that would eliminate high frequencies which are caused by nonlinearitie s and especially shocks. A variety of numerical algorithms and computer codes for the Euler equations have been developed. Methods such as MacCormack's1 explicit scheme and Steger's2 application of the Beam and Warming3 implicit algorithm are in wide use. Some notable recent developments based on explicit Runge-Kutta schemes are the work of Jameson et al.4 and Rizzi and Eriksson.5 The time integration scheme, boundary condition treatment, and other details are different from method to method. These all have one thing in common: The use of a basically central difference approxima- tion to the spatial derivatives and the addition of some form of artificial dissipation. In contrast there is a currently popular class of schemes, (monotone, total variation diminishing (TVD), flux split, flux difference, lambda) that employ some form of upwind differencing under the assumptions of characteristic theory and wave propagation. The work of Steger and Warming,6 Roe,7 Van Leer,8 Osher and Chak- ravarthy,9 and Marten's TVD methods10 all fall in this category. Although it will not be shown here for every case, these schemes are all equivalent to a central differencing scheme plus some form of dissipation. The addition of artificial dissipation to central differenc- ing will be the focus of this paper. It is added for two main reasons: first, to control the odd-even uncoupling of grid points typical of central differencing and, second, to control strong nonlinear effects such as shocks. The paper in- vestigates various forms of artificial dissipation employed in the Euler codes. The particular difference forms and theory behind the choice of coefficient will be examined here, as well as some stability and accuracy arguments when possible. Results for flow over airfoils with shocks will be used as test cases. A recent version of Steger's 2 two-dimensional im-