A new approach based on the method of characteristics (MOC) and Rosenbrock method is developed to solve the time-dependent transport equation in one-dimensional (1D) geometry without any approximation and considering delayed neutrons. Within the MOC methodology, the leakage term in time-dependent transport equation can be simplified to spatial derivative of the angular flux along the characteristics lines. For 1D geometry, the proposed exponential correlation derived from the steady-state MOC equations provides the correlation between the cell outgoing angular flux and the cell average angular flux. Thus, the spatial derivative term can be further substituted by the relation containing only the cell average angular flux that represents the unknowns. Therefore, the 1D time-dependent transport equation is decomposed into a series of locally coupled ordinary differential equations (ODE). Rosenbrock method was chosen to solve the system of ODEs. It is a fourth order explicit method with automatic step size control feature developed for stiff ODEs. The FORTRAN90 numerical program is developed to thus solve the time-dependent transport equation considering delayed neutrons in 1D geometry with both vacuum and reflective boundary conditions. The step perturbation is currently supported. The method presented in this paper was verified in comparison to 1D fast reactor benchmark showing good accuracy and efficiency.
An Adaptive Algorithm for the Time Dependent Transport Equation with Anisotropic Finite Elements and the Crank–Nicolson Scheme
