Bicompact Rogov schemes for the multidimensional inhomogeneous linear transport equation at large optical depths

Abstract

Bicompact Rogov schemes intended for the numerical solution of the inhomogeneous transport equation are extended to the multidimensional case. A factorized modification of the method without using splitting in directions or introducing additional half-integer spatial points is proposed. As its original counterpart, the scheme is fourth-order accurate in space and third-order accurate in time. In the case of one dimension, a higher order accurate scheme on a minimal stencil is constructed using the node values of the unknown function and, in addition, its integral averages over a spatial cell. In the case of two dimensions, the set of unknowns in a given cell is expanded to four. The resulting system of equations is solved for the expanded set of variables by the running calculation method, which reflects the characteristic properties of the transport equation without explicit use of characteristics. In the case of large optical depths and a piecewise differentiable solution, a monotonization procedure is proposed based on the Rosenbrock scheme with complex coefficients.