Temporal accuracy analysis of phase change convection simulations using the JFNK-SIMPLE algorithm

The incompressible Navier-Stokes and energy conservation equations with phase change effects are applied to two benchmark problems: (1) non-dimensional freezing with convection; and (2) pure gallium melting. Using a Jacobian-free Newton-Krylov (JFNK) fully implicit solution method preconditioned with the SIMPLE (Numerical Heat Transfer and Fluid Flow. Hemisphere: New York, 1980) algorithm using centred discretization in space and three-level discretization in time converges with second-order accuracy for these problems. In the case of non-dimensional freezing, the temporal accuracy is sensitive to the choice of velocity attenuation parameter. By comparing to solutions with first-order backward Euler discretization in time, it is shown that the second-order accuracy in time is required to resolve the fine-scale convection structure during early gallium melting. Qualitative discrepancies develop over time for both the first-order temporal discretized simulation using the JFNK-SIMPLE algorithm that converges the nonlinearities and a SIMPLE-based algorithm that converges to a more common mass balance condition. The discrepancies in the JFNK-SIMPLE simulations using only first-order rather than second-order accurate temporal discretization for a given time step size appear to be offset in time.