Stability of convective flows in cavities: Solution of benchmark problems by a low-order finite volume method

A problem of stability of steady convective flows in rectangular cavities is revisited and studied by a second-order finite volume method. The study is motivated by further applications of the finite volume-based stability solver to more complicated applied problems, which needs an estimate of convergence of critical parameters. It is shown that for low-order methods the quantitatively correct stability results for the problems considered can be obtained only on grids having more than 100 nodes in the shortest direction, and that the results of calculations using uniform grids can be significantly improved by the Richardson's extrapolation. It is shown also that grid stretching can significantly improve the convergence, however sometimes can lead to its slowdown. It is argued that due to the sparseness of the Jacobian matrix and its large dimension it can be effective to combine Arnoldi iteration with direct sparse solvers instead of traditional Krylov-subspace-based iteration techniques. The same replacement in the Newton steady-state solver also yields a robust numerical process, however, it cannot be as effective as modern preconditioned Krylov-subspace-based iterative solvers.