Fractional Bernstein polynomial method for solving time-fractional neutron diffusion systems

In this research, an innovative numerical method is presented to solve the fractional space-time multigroup neutron diffusion model. The method employs the fractional-order Bernstein polynomials for temporal calculations and higher order finite difference schemes for spatial discretization. Two major drawbacks should be addressed to solve such fractional diffusion model. The first is that the spatial discretization produces a large system of ordinary differential equations with a fundamental square matrix of huge size. To address this problem, the used numerical method should be simple and fast to overcome such difficulty. Therefore, a straightforward recurrence relation for the Bernstein coefficient estimations is introduced. The second is that the fractional operator consumes more computational times due to its persistent memory, especially when using smaller values of the anomalous diffusion exponents. This problem is overcome by employing the technique of automatic time step size control to reduce the number of iterations and CPU times. Also, without causing significant change in the system complexity, a higher order central finite difference scheme with accuracy order equals 9(Delta 4) is employed for spatial discretization of the core interior mesh points. To further improve the stationary state results, a new scheme with accuracy order equals 9(Delta 3) is introduced using seven point stencils at the core boundaries. It is proved that using these higher order schemes at both the interior and boundary mesh points significantly improves the accuracy of the steady state calculations. The method is applied to homogenous and heterogonous benchmark reactors with varying anomalous subdiffusion exponents. Numerical simulation of assembly power densities and temperature distributions for LRA BWR reactor including adiabatic heating and Doppler feedback is presented. It is proved that for heterogeneous reactors, the effects of the fractional orders diminish throughout the transient for step, ramp perturbations and also with the presence of thermal-hydraulic temperature feedback.