Monotonization of a modified scheme with Hermitian interpolation for the numerical solving of an inhomogeneous transport equation with absorption term

A hybrid scheme for the numerical solving of a transport equation is implemented. The high order scheme is a modified CIP (Cubic Interpolation Polynomial) scheme with Hermitian interpolation. The third order of approximation of the CIP scheme is achieved by including in the list of unknowns not only the nodal values of the function, but also the nodal values of its derivatives. In the modification under consideration the Euler–Maclaurin formula is used to calculate derivatives. The low order scheme is the characteristic scheme of the first order of approximation. Local, layerwise and global monotonization are considered, the hybridization is performed, respectively, after a cell, a time layer or the whole grid processing. It is shown that the best results are obtained by a scheme with local monotonization. The convergence orders of the hybrid scheme on tests of different smoothness of exact solution do not differ significantly from the convergence orders of the CIP scheme. It is proposed to calculate an integral along a characteristic using the Simpson formula for the Stieltjes integral in the case of the large optical thickness; it provides a significant reduction in the numerical solution errors.