Higher orders of Magnus expansion for point kinetics telegraph model

Abstract

Neutron transport has been still under very active development in research institutions and academia throughout the world. The spatial, temporal, energy and directional angle dependence make it remains one of the most computationally challenging problems in the world. The correct P1 approximation to the neutron transport equation is not the first order diffusion equation, but the second-order in time, which is called telegraph equation. In this paper, a synopsis derivation of the point kinetics telegraph model is obtained from the neutron transport equation as a couple system of stiff differential equations. The problem of the obtained system is formulated in the matrix form and solved by higher orders Magnus expansion, where the first successive three orders of Magnus expansion are derived analytically for the point telegraph equations with multi-group of delayed neutrons and various different types of reactivity. These approximations depend on the exponential function of Magnus matrix, where the calculations are obtained using the eigenvalues of Magnus matrix and the corresponding eigenvectors. The proposed methods are tested using different cases of reactivity such as step, ramp and sinusoidal reactivity insertions. The numerical results indicate that the high order of Magnus expansion approximations is accurate compared with the traditional methods.