Solution of the multidimensional compressible Navier-Stokes equations by a generalized implicit method

Abstract In an effort to exploit the favorable stability properties of implicit methods and thereby increase computational efficiency by taking large time steps, an implicit finite-difference method for the multidimensional Navier-Stokes. equations is presented. The method consists of a generalized implicit scheme which has been linearized by Taylor expansion about the solution at the known time level to produce a set of coupled linear difference equations which are valid for a given time step. To solve these difference equations, the Douglas-Gunn procedure for generating alternating-direction implicit (ADI) schemes as perturbations of fundamental implicit difference schemes is employed. The resulting sequence of narrow block-banded systems can be solved efficiently by standard block-elimination methods. The method is a one-step method, as opposed to a predictor-corrector method, and requires no iteration to compute the solution for a single time step. Test calculations are presented for a three-dimensional application to subsonic flow in a straight duct with rectangular cross section. Stability is demonstrated for time steps which are orders of magnitude larger than the maximum allowable time step for conditionally stable methods as determined by the well-known CFL condition. The computational effort per time step is discussed and is very approximately only twice that of most explicit methods. The accuracy of computed solutions is examined by mesh refinement and comparison with other analytical and experimental results.

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