ICASE Report No . 90-54 ICASE MULTIGRID FOR HYPERSONIC INVISCID FLOWS

We consider the use of multigrid methods to solve the Euler equations for hypersonic flow. We consider the steady state equations with a Runge-Kutta smoother based on the time accurate equations together with local time stepping and residual smoothing. We examine the effect of the Runge-Kutta coefficients on the convergence rate considering both damping characteristics and convection properties. We also show the importance of boundary conditions on the convergence rate for hypersonic flow. Also of importance are the switch between the second and fourth difference viscosity. Solutions are given for flow around a bump in a channel and flow around a biconic section. This research was partially supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-18605 while the authors were in residence at ICASE, NASA Langley Research Center, Hampton, Va. Introduction. Central difference type explicit schemes are currently being applied on a regular basis in the solution of the Euler and Navier-Stokes equations see, e.g [171. In these applications multigrid is used to accelerate the basic explicit schemes. However, in most cases the applications have been to flows in the transonic or low supersonic regime. Higher speeds have generally been treated without multigrid or using semi-coarsening [8, 10]. Multigrid methods were first developed for elliptic equations. These were later extended to hyperbolic equations such as the fluid dynamic equations for subsonic and transonic flow [4, 7, 12]. For these cases the steady state retains many of the properties of an elliptic equation since the region of supersonic flow is limited. We shall show that with proper care the multigrid method still works for hypersonic flow. Gustafsson and Lotstedt [2] have pointed out that hyperbolic multigrid works by two different processes. For the long waves the advection process is most important and multigrid achieves its efficiency by allowing the use of larger time steps on coarser grids. Hence, it is important that the smoother use large time steps. However, for the shorter waves dissipation is more important and the efficiency of multigrid is based on principles similar to that for elliptic equations. In this study we consider a Runge-Kutta scheme [3] as the smoother for the multigrid method. The central differences are augmented by an artificial viscosity based on TVD principles [13]. A number of changes are made to the scheme to enable it to work for higher speed flows. We consider both unbounded domains and domains with solid surfaces at the boundaries. The main difficulties are not due to the basic multigrid method but rather to the imposition of boundary conditions. We first describe the basic Runge-Kutta method for the central difference scheme together with a description of the artificial viscosity. We then describe the theory of the multigrid acceleration for a simple advection equation. We finally present several cxamples to demonstrate our conclusions. Basic Scheme The basic elements of the scalar dissipation model considered in this paper were first :. r. 'r~ced by Jameson, Schmidt, and Turkel [3] in conjunction with Runge-Kutta exsce.es. Te space discretization is based on central differences with an additional ar. C cosi:v. This algorithm has been used by many investigators to numerically . o ""e BLer ecaions for a wide range of fluid dynamic applications. The same type -"-ae :reza::c. : nas been applied to alternating direction implicit (ADI) schemes E. :-. :c:: sc-. emes '5*. In this section the basic scheme is briefly -. e -. -e e . s :r. *ne :orrn S =0, . , " z: :-': :, " v r c o.served variables, and f,g are the flux . azr " are -. e 4 ar.d Cartesian coordinates (x,y). If (1) • . " . ,. :r. :r _ . a res = 'z . ' and r xy , then w e