The application of a finite volume multigrid method to three-dimensional flow problems in a highly viscous fluid with a variable viscosity

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Abstract
In this paper we discuss the application of a finite-volume multigrid method to solve three-dimensional thermally driven convection in a highly viscous, incompressible fluid with a variable viscosity. The conservation laws are solved in the primitive variable formulation, A second-order control volume method is used as discretization. Two schemes are used for time stepping, a second-order implicit-explicit scheme based on the Crank-Nicolson and Adams-Bashforth method, and a fully-implicit θ-method. The implicit system of nonlinear equations are solved using multigrid iteration with the SIMPLER method as smoother. In this paper, we describe the implemented multigrid method and investigate its efficiency and the robustness for different viscosity contrasts. Convergence tests showed that with a small modification of the SIMPLER method, the multigrid method exhibits a satisfactory convergence rate even for viscosity contrasts up to 109. Three cases of time-dependent thermally driven convection with viscosity contrasts up to 105 are considered and discussed and the multigrid method has demonstrated its robustness also for these cases. Further, we have also compared the computational efficiency of the two time stepping methods. It appeared that the fully-implicit scheme is a little more efficient than the implicit-explicit scheme for a constant viscosity and it was considerably more efficient for a viscosity contrast of 103