A spatially-factored fully implicit solution method for block structured meshes with irregular nodes

Abstract

The alternating direction multi-zone implicit (ADMZI) method for solving implicitly structured multi-zone domains by spatial factorization techniques is extended to block structured domains that include irregular nodes, i.e. nodes that have a different connectivity with the neighboring nodes. It is shown for fully implicit schemes that the presence of irregular nodes (in the two-dimensional case) imposes the use of a mixed stage in the ADI factorization, in addition to the two standard stages. In the mixed stage, there are parts where the equations of the first stage are solved, while in the other parts the first stage has been previously completed and the equations of the second stage are being solved. It is proven that the overall second-order temporal accuracy of the solution is preserved. Generalized ADMZI sweeps are defined to allow a consistent and unified approach to all the ADMZI factorization stages, including the mixed stage. This novel methodology can be employed for solving implicitly many existing solvers of systems of partial differential equations, such as the Navier–Stokes equations, using spatially factored techniques and block structured meshes with irregular nodes. The resulting scheme is efficient because the computational work is equivalent to a single iteration of most other existing iterative solvers.