Characteristic Scheme for the Solution of the Transport Equation on an Unstructured Grid with Barycentric Interpolation

Abstract

An interpolative-characteristic method is constructed with the approximation not lower than the second order for solving the transport equation on an unstructured grid composed of tetrahedra. The problem of finding a numerical solution by this method, further referred to as the method of short characteristics, is subdivided into two subproblems. The first concerns the resolution of a single simplicial cell. A set of grid values must be specified, which when set on the illuminated faces is mathematically sufficient for obtaining all the remaining grid values in the cell. Depending on the location of the cell and the propagation direction of the radiation there are three different types of illumination. It is proposed to use interpolation of a cell in barycentric coordinates with 14 free coefficients, which enables taking into account the values of the radiation intensity at the nodes and the average integral values of the intensity over the edges and faces without adding new stencil points. This interpolation ensures at least the second order of approximation with additional allowance for the terms with the third-order approximation. Moreover, the method takes into account a conservative redistribution of the outbound flow over the edges of the cell. The second subproblem is associated with choosing the order of tracing the cell and can be solved using methods of graph theory. The numerical calculations confirmed approximately the second order of convergence.