Abstract <p>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 joint system of kinetic equations of high (HO) and low (LO) orders. The monotonization algorithm for the quasi-diffusion method from the HOLO algorithm family is used in the work. It does not include an iterative process, therefore, it is time-efficient. Bicompact schemes are constructed using the method of lines within a single cell. The schemes 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 the system of low-dimensional equations leads to a decrease in the convergence order in time of the Runge–Kutta method under consideration to the second order. Thus, the orders of convergence in time for both methods are the same. The properties of the schemes are studied in application to a nonstationary generalization of the model problem of neutron transport (Reed problem). It is demonstrated that the monotonization algorithm is efficient for the methods 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 order.</p>