The use of domain decomposition in accelerating the convergence of quasihyperbolic systems

This paper proposes an alternate form of the active-domain method [K. Nakahashi and E. Saitoh, AIAA J. 35, 1280 (1997)] that is applicable to streamwise separated flows. Named the "marching window," the algorithm consists of performing pseudo-time iterations on a minimal width subdomain composed of a sequence of cross-stream planes of nodes. The upstream boundary of the subdomain is positioned such that all nodes upstream exhibit a residual smaller than the user-specified convergence threshold. The advancement of the downstream boundary follows the advancement of the upstream boundary, except in zones of significant streamwise ellipticity, where a streamwise ellipticity sensor ensures its continuous progress. Compared to the standard pseudo-time-marching approach, the marching window decreases the work required for convergence by up to 24 times for flows with little streamwise ellipticity and by up to eight times for flows with large streamwise separated regions. Storage is reduced by up to six times by not allocating memory to the nodes not included in the computational subdomain. The marching window satisfies the same convergence criterion as the standard pseudo-time-stepping methods, hence resulting in the same converged solution within the tolerance of the user-specified convergence threshold. The algorithm is not restricted to a discretization stencil and pseudo-time-stepping scheme in particular and is used here with the Yee-Roe scheme and block-implicit approximate factorization solving the Favre-averaged Navier-Stokes (FANS) equations closed by the Wilcox kω turbulence model. The eigenstructure of the FANS equations is also presented.