Numerical simulation of an anisotropic heat transfer in magnetized neutron stars with 3D basic operators method

Abstract

We have solved numerically a three dimensional boundary-value problem for a heat transfer equation in a magnetized neutron star crust with an updated tensorial heat conductivity coefficient. The temperature distribution in a neutron star crust in presence of a magnetic field was simulated. To calculate the surface temperature distribution, we have constructed a local one-dimensional plane-parallel model of a magnetized outer envelope of the neutron star and used it as an outer boundary condition for 3D problem to find a self-consistent solution.This problem was solved with our extension of a basic (support) operators numerical method on a tetrahedral mesh. The idea of operator approach consists of inclusion the boundary conditions into difference form of the solving problem and its formulation as one operator equation. The finite difference operators are constructed in the way to fulfill corresponding relations between continuous operators (for example, div(curl)=0, div is conjugated to -grad so div(grad) is self-conjugated etc.). Such approach allows to obtain completely conservative implicit finite difference schemes. Efficient iterative methods can be used to find the solution because constructed matrixes have good properties, such as symmetry and positive definiteness.