Application of the Multidomain Local Fourier Method for CFD in Complex Geometries

Abstract

A low communication parallel algorithm is developed for the solution of time-dependent non-linear PDE’s. Our particular interest is in the application of this algorithm to fluid dynamics problems. The parallelization is achieved by domain decomposition. The discretization in time is performed via a third order semi-implicit stiffly stable scheme. The elemental solutions in subdomains are constructed by using a high-order method with the Local Fourier Basis (LFB). It results in elliptic equations of Helmholtz and Poisson types, which have to be solved repeatedly at each time step. The continuity of the global solution is accomplished using a point-wise matching of the local subsolutions on the interfaces. The matching relations are derived in terms of the jumps on the interfaces. The LFB transformation enables the splitting a 2-D problem with global coupling of the interface unknowns into a set of uncoupled 1-D differential equations. Localization properties of an elliptic operator, resulting from the discretization in time of a time-dependent problem, are utilized in order to simplify the matching relations. In effect, only local (neighbor-to-neighbor) communication between the processors becomes necessary. The present method allows the treatment of problems in various complex geometries by the mapping of curvilinear domains into simpler (rectangular or circular) regions with subsequent matching of local solutions. The operator with non constant coefficients, obtained in the transformed domain, is preconditioned by an appropriate constant coefficient operator, easily inverted by the LFB. The problem is then solved with spectral accuracy by (a rapidly convergent) conjugate gradient iteration.1