The Alternating Direction Multi-zone Implicit Method

Abstract

Within the structured grid approach, numerical solution of partial differential equations (PDE) in complex regions requires the decomposition of the domain into several zones. Implicit solution methods of the discrete equations are preferred because of their superior numerical properties. Indeed, many existing multi-zone solution methods use implicit techniques, but the zonal boundaries are updated explicitly. The zonal boundaries are created between the zones in the process of domain decomposition, but otherwise they are regular interior field points. If only steady state solutions are sought, the explicit calculation of the zonal boundaries usually effects only the convergence rate but not the accuracy of the solution. However, in time-dependent cases this may degrade the time accuracy of the solution. In the present work, we propose a novel fully implicit method for solving sets of PDE using multi-zones and structured grids. The method combines the zonal approach with the alternating direction implicit (ADI) method, and hence the method is referred to as the alternating direction multi-zone implicit (ADMZI) method. The key idea is the generation of different sets of zones for each stage (factor) of the ADI method, rather than using the same set of zones for all the stages. Consequently, the ADI sweeps are performed between physical boundaries, so that zonal boundaries are avoided and the solution is fully implicit. The ADMZI method can be applied to any set of PDE that employs an ADI (or approximate factorization) method. Typical examples include the time-dependent compressible or incompressible Navier-Stokes equations. Several conceptual examples and actual numerical test cases (that solve the heat conduction equation) confirm the versatility, efficiency, and accuracy of the ADMZI method.