Non-iterative monotonization of high-order bicompact schemes in the quasi-diffusion method of solving the transport equation

An improved algorithm for monotonization of bicompact schemes for the HOLO algorithm for solving the transport equation is proposed and implemented. HOLO algorithms allow to accelerate the convergence of iterations when solving interconnected systems of kinetic equations of high (HO) and low (LO) orders. The monotonization algorithm used in the work for the quasi-diffusion method from the HOLO algorithm family does not include an iterative process, therefore it is time-efficient. Bicompact schemes are constructed using the method of lines within a single cell, have the fourth order of approximation in space, and can be integrated over time using various methods. The Runge-Kutta method of the third order of approximation and the trapezoid method of the second order of approximation are considered as methods of integration over time. The implementation of classical boundary conditions in the quasi-diffusion method for a system of low-dimensional equations leads to a decrease in the convergence order in time to the second in the Runge-Kutta method under consideration, so the orders of convergence in time for both methods are the same. The properties of the schemes are investigated in application to a non-stationary generalization of the model problem of neutron transport (Reed problem). It is demonstrated that the monotonization algorithm is efficient if the considered methods are used for time integration. It is shown that the Runge-Kutta method of the third order of approximation is more time-efficient and reliable, despite the fact that due to the boundary conditions the order of convergence is reduced to the second.