A semi-implicit unstructured operator-difference scheme for three-dimensional self-gravitating flows
A support operators (operator-difference) method has proven itself well for implicit simulations of different astrophysical fluid flows. Following the operator-difference approach, we construct nodal difference analogues of differential operators in Cartesian coordinates, where the conjugacy properties are the same as for original ones. Using the difference operators, we develop an Eulerian semi-implicit gas-dynamical solver with self-gravity on a three-dimensional collocated unstructured tetrahedral mesh. In the solver, only acoustic waves are treated implicitly, resulting to the only elliptic equation for a pressure on each time-step. The conjugacy properties of derived difference operators allow us to construct symmetric sign-definite matrices for this elliptic equation as well as for Poisson equation for a gravitational potential. The stability condition of the proposed scheme is milder, than the usual Courant-Friedrichs-Lewy condition for explicit solvers, and depends only on the gas velocity. Results of test problems’ simulations of low and high Mach number flows are presented.