An Implicit Factored Scheme for the Compressible Navier-Stokes Equations

TLDR

An implicit finite‑difference scheme is developed to solve the compressible Navier–Stokes equations in conservation‑law form. The scheme employs explicit evaluation of spatial cross derivatives and is formulated in terms of conserved‑variable and flux‑increment equations to enable a factored, efficient algorithm. It attains second‑order time accuracy, is noniterative, unconditionally stable, requires only two time‑level data storage, and its delta form yields steady‑state solutions independent of time step, as shown on a two‑dimensional shock–boundary‑layer interaction test.

Abstract

An implicit finite-difference scheme is developed for the numerical solution of the compressible Navier-Stokes equations in conservation- law form. The algorithm is second-order- time accurate, noniterative, and spatially factored. In order to obtain an efficient factored algorithm, the spatial cross derivatives are evaluated explicitly. However, the algorithm is unconditional ly stable and, although a three-time-lev el scheme, requires only two time levels of data storage. The algorithm is constructed in a form (i.e., increments of the conserved variables and fluxes) that provides a direct derivation of the scheme and leads to an efficient computational algorithm. In addition, the delta form has the advantageous property of a steady state (if one exists) independent of the size of the time step. Numerical results are presented for a two-dimensiona l shock boundary-layer interaction problem.