Solution of the three-dimensional compressible Navier-Stokes equations by an implicit technique

Abstract

One of the major obstacles to the solution of the multidimensional compressible Navier-Stokes equations is the large amount of computer time generally required, and consequently, efficient computational methods are highly desirable. Most previous methods have been based on explicit difference schemes for the unsteady form of the governing equations and are subject to one or more stability restrictions on the size of the time step relative to the spatial mesh size. These stability limits usually correspond to the well-known Couraut-Friedrichs-Lewy (CFL) condition aud~ in some schemes, to an additional stability condition arising from viscous terms. These stability restrictions can lower computational efficiency by imposing a smaller time step than would otherwise be desirable. Thus, a key disadvantage of explicit methods subject to stability limits is that the maximum time step is fixed by the spatial mesh size rather than the physical time dependence or the desired temporal accuracy. In contrast to explicit methods, implicit methods tend to be stable for large time steps and hence offer the prospect of substantial increases in computational efficiency, provided of course that the computational effort per time step is competitive with that of explicit methods. Iu an effort to exploit these favorable stability properties~ an implicit method based on alternating-direction differencing techniques was developed and is discussed herein. The present method can be briefly outlined as follows: The governing equations are replaced by either a Crauk-Nicolsou or backwsrd time difference approximation. Terms involving nonlinearities at the implicit time level are linearized by Taylor expansion about the known time level, and spatial difference approximations are introduced. The result is a system of multidimen- sional coupled (but linear) difference equations for the dependent variables at the unknown or implicit time level. To solve these difference equations, the Douglas-Guuu (Ref. 1 ) procedure for generating alternating-direction implicit (ADI) schemes as perturbations of fundamental implicit difference schemes is introduced. This technique leads to systems'of one-dimensional coupled linear difference equations which can be solved efficiently by standard block-elimination methods. Complete details of the method are given in Ref. 2.