Application of 3D basic operators method extension to astrophysical problems

Abstract

Basic (support) operators method (Samarskii’s method, operator-difference method) has proven itself well in 2D numerical simulations of the astrophysical problems. An idea of the operator approach consists of the inclusion of boundary conditions in a finite difference form into the grid analogue of solving problem and formulation of the finite difference problem as an operator equation. The finite difference operators are constructed in the way to fulfill corresponding relations between continuous operators (for instance, div(rot)=0, div is conjugated to -grad; -div(grad) is self-conjugated etc.). The approach allows obtaining completely conservative finite difference schemes. The matrix which corresponds to the self-conjugated operator is symmetrical and can be inversed efficiently by modern iteration methods. In the paper a 3D generalization of 2D grid analogues for continuous differential operators using a cell-node approximation on triangular grid was realized for tetrahedral mesh. For testing we calculated a 3D Newtonian gravitational potential, and a stationary heat transfer equation in spherical layer with the first type boundary condition on the inner surface and the third type boundary condition on the outer one was calculated. The method was applied to simulation of anisotropic heat transfer in magnetized neutron star crust.