Euler equations - Implicit schemes and boundary conditions

Implicit boundary condition procedures are presented for use with implicit finite difference schemes for the unsteady Euler equations. This new boundary point treatment is based on the mathematical theory of characteristics for hyperbolic systems of equations. Along with the theoretical background, the practical application of the method to several types of boundaries is also explained using several examples. The specific boundary conditions covered include subsonic inflow and outflow, surface tangency, and shock waves. The example problems include one-dimensional Laval nozzle flow, dual-throat rocket engine nozzle flow, and supersonic flow past a sphere. The implicit boundary treatment permits the use of large time steps allowing the finite difference algorithm to converge to the asymptotic steady state much faster than schemes that use explicitly applied boundary conditions. At least an order of magnitude increase in computational speed is demonstrated in the examples shown.' Background T HE growing popularity of solutions to the Euler equations in transonics and their continued application in supersonics have increased the need for quicker solutions. The potential of implicit schemes in this direction has not been fully exploited for want of correct, implicit application of boundary conditions. The predominant use of implicit algorithms for the Navier-Stokes equations has partly been responsible for the neglect of implicit boundary point treatment for the Euler equations. Thus, there is a need for correct and stable procedures for the easy implicit application of boundary conditions. Such methods will serve the two purposes of 1) reaching time-asymptotic steady state faster and 2) permitting a time step for truly unsteady flow that is not necessarily restricted by the CFL stability criterion but is based upon the magnitude of the transients. For clues and information on how to construct such boundary condition procedures, one must turn to the mathematical theory of characteristi cs for hyperbolic systems of equations. The unsteady Euler equations belong to this category. The theory for hyperbolic systems is rich with information on signal propagation directions. The characteristics theory clearly points to the number of boundary conditions that may and need be prescribed without overdetermining the solution. Boundary condition procedures based on this theory have been known and applied for several years by Kentzer, 1 Porter and Coakley,2 de Neef, 3 and others. In earlier work by this author,4'5 easily understood and implementable methods for boundary point treatment were presented. However, all of the above techniques were developed for explicit finite difference schemes. It seems that it must be easy to extend such methodologies based on mathematical theory for hyperbolic systems to implicit finite difference schemes, and indeed, it is simple enough. The rest of this paper describes such implicit boundary condition procedures. The given examples illustrate in detail the application of the proposed methodology to specific types of boundaries and demonstrate the merits of the new scheme.