The momentum interpolation method based on the time-marching algorithm for All-Speed flows

The time-marching approach has clear physical meaning and strict mathematical nature and has been applied in computation of compressible flows widely and extended to many uniform algorithms for All-Speed flows. Remedy for its weakness in the problem of checkerboard decoupling of pressure field for incompressible flows is envisaged with the time-marching momentum interpolation method (MIM) taken into account in this paper. Existing preconditioning methods for suppressing decoupling and time-marching MIM are analyzed for this purpose, and algorithms of time-marching MIM are proposed for steady and unsteady flows and for All-Speed flows. Asymptotic analysis shows that the supposed time-marching MIM has at least a third-order accuracy, better than the existing time-marching coupling methods, which only have an accuracy of the same order as the adopted scheme has. Effects of the time step sizes on the ability of the time-marching MIM to suppress the checkerboard pressure decoupling are particularly discussed in terms of the dual-time stepping approach, and it is revealed how the decreased sizes of either the pseudo- or physical-time step increases the possibility of decoupling and how Choi’s modification, in which the history of the interface velocity is decided by itself instead of the arithmetic average of the velocities on its adjacent nodes, removes the unphysical pressure oscillation with small size of the physical time step but leads to divergence with the pseudo-time step as well. As a remedy for the pseudo-time step, such methods are recommended as implicit methods and the local-time step method with a proposed modification of the time-marching MIM preventing accuracy loss due to very large time step size. Numerical experiments support the theoretical analyses and show the validity of the time-marching MIM proposed.