An Implicit Numerical Method for the Multidimensional Compressible Navier-Stokes Equations.

TL;DR: 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, and is examined in a three-dimensional application to subsonic flow in a straight duct with rectangular cross section.

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 is based on a fully-implicit backward time difference scheme which is linearized by Taylor expansion about 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 introduced. The resulting sequence of one-dimensional equations 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. The stability and accuracy of the method are examined in a three-dimensional application to subsonic flow in a straight duct with rectangular cross section. (Modified author abstract)