A switching ILU(0)-SGS preconditioner for matrix systems of incompressible flow and heat transfer using condition number estimates

Preconditioning of matrix systems obtained from an unstructured finite volume discretisation of the incompressible NS equations are studied. The condition number is first estimated by providing Gershgorin-type theoretical bounds. The linear algebraic equations are solved using the preconditioned-BiCGSTAB algorithm. Condition numbers of velocity and temperature matrices show these to be well-conditioned compared to pressure. As a result, the symmetric Gauss Seidel (SGS) preconditioner performs well for velocity matrices on single and multiple processors when compared to ILU(0) that requires LU factorisation. The present study proposes a preconditioning algorithm that switches from ILU(0) to the SGS preconditioner for velocity and temperature. The composite algorithm shows a reduction in the overall simulation time. For a power law fluid, the upper bound of the condition number correlates with the power law index. For flow past a circular cylinder, the temporal oscillation of the largest singular value relates to the onset of vortex shedding.