Solving the Time-dependent Transport Equation Using Time- Dependent Method of Characteristics and Rosenbrock Method

A new approach based on the method of characteristics (MOC) and Rosenbrock method is developed to solve the timedependent 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 timedependent 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. INTRODUCTION The time-dependent transport modeling for arbitrary reactor geometry is important field in kinetics analysis. Such modeling requires solving the time-dependent transport equation with delayed neutrons. In this paper, we present the coupling of the MOC and Rosenbrock to solve 1D timedependent neutron transport equation. METHODOLOGY In order to be able to solve the time-dependent neutron transport equation, we need to discretize the complex partial differential equations (PDE) into ordinary differential equations (ODE); then, an appropriate numerical method can be selected to integrate the system of ODE over the time. In the MOC the leakage term can be simplified and the method does not pose any limit in describing and therefore modeling neutron transport in multidimensional arbitrary geometries. The following is the derivation of the equations that starts from the discretized 1D, multi-group MOC transport equations:            