Bicompact scheme for the multidimensional stationary linear transport equation

A fourth-order accurate (in space) bicompact scheme is proposed for solving the inhomogeneous stationary transport equation in two dimensions. The scheme is based on a minimal stencil consisting of two nodes in each dimension and is obtained as a stationary limit of bicompact schemes produced by the method of lines for the nonstationary transport equation. The set of unknowns for each two-dimensional cell consists of the node values of the solution function and its integrals over cell edges and the entire cell. A closed system of linear equations is obtained for all desired variables in each cell. This system is solved using the running calculation method, which reveals the characteristic properties of the transport equation without explicitly using characteristics. The numerical results are compared with the solution produced by a conservative-characteristic method applied to a similar set of variables. The advantages of the bicompact schemes are demonstrated.