Domain Decomposition Method Usage for Parallelization and Extra Preconditioning of Viscous Incompressible Flow Simulation by Using the LS-STAG Method

As a rule, the main part of the computational costs in the numerical solution of problems in continuum mechanics consists in the solving of large sparse systems of linear algebraic equations. For this reason, efficient parallelization of this particular procedure can significantly speed up the simulation. To solve this problem, two main approaches can be used. The simplest approach consists in parallelizing of matrix-vector operations in a usual iterative solver. It requires several synchronization points and exchanges of coefficients at each iteration of the solver, which does not significantly speed up the simulation process as a whole. Therefore, domain decomposition methods are preferable. These methods involve dividing the computational domain into subdomains, constructing and solving separate problems in them, as well as some procedure to coordinate the solution between subdomains to ensure global convergence. Subdomains can overlap, as in the Schwartz method used in OpenFOAM, or they can be separated by interface sections, on which their own interface task is built, as in the Schur complement method. The latter method is used in this research to construct a parallel algorithm for viscous incompressible flow simulation by using the immersed boundary method LS-STAG with cut-cells and level-set functions. The resulting matrix of the interface system has a block tridiagonal structure. To speed up prototyping, OpenMP parallel programming technology is used in the software implementation of the developed algorithm, so computational experiments are carried out only on systems with shared memory, in particular on individual nodes of the educational and experimental cluster of the Applied Mathematics Department, Bauman Moscow State Technical University. To verify and evaluate the effectiveness of the developed parallel algorithm, a well-studied test problem about simulation of two-dimensional flow around a stationary circular airfoil is considered. Computations on a sequence of meshes with different numbers of subdomains show that the parallel algorithm allows one to obtain the same numerical solution as the original algorithm of the LS-STAG method, and the computed values of the Strouhal number and drag coefficient are in good agreement with the experimental and computational data known in the literature. Experiments demonstrate that the developed algorithm with domain decomposition allows to accelerate simulation even in sequential mode by reducing the number of solver iterations, i.e. the domain decomposition method acts as an additional preconditioner. Due to this property, the acceleration is superlinear when simulating in parallel mode with developed algorithm. This effect persists up to a certain number of subdomains, which depends on the size of the problem.

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