Numerical Schemes for Fluid Flow and Heat Transfer in Medical Simulations

Abstract

Incompressible fluid flow is governed by the Navier-Stokes equation, which, together with the diffusion and continuity equations, forms a coupled system of partial differential equations that have to be solved to simulate the fluid dynamics. We describe a finite difference scheme and boundary conditions used to solve the system of partial differential equations on general 3-dimensional domains with explicit integration in time. The most computationally intensive part is the pressure equation that requires the solution of a sparse linear system in each time-step of the simulation. Various iterative methods for the solution of the linear system are tested and compared among which the multigrid method outperforms others. Some test examples are given to prove the validity of the simulation results. The paper concludes with an analysis of parallel computational complexity of the SOR method and parallelization strategy for the multigrid method.