A Taylor series solution of the reactor point kinetics equations

The method of Taylor series expansion is used to develop a numerical solution to the reactor point kinetics equations. It is shown that taking a first order expansion of the neutron density and precursor concentrations at each time step gives results that are comparable to those obtained using other popular and more complicated methods. The algorithm developed using a Taylor series expansion is simple, completely transparent, and highly accurate. The procedure is tested using a variety of initial conditions and input data, including step reactivity, ramp reactivity, sinusoidal, and zigzag reactivity. These results are compared to those obtained using other methods.

💡 Research Summary

The paper introduces a straightforward numerical method for solving the reactor point‑kinetics equations by employing a first‑order Taylor series expansion at each time step. The point‑kinetics model consists of a coupled set of ordinary differential equations describing the time evolution of the neutron density n(t) and the concentrations of several delayed‑neutron precursor groups C_i(t) under a prescribed reactivity insertion ρ(t). Traditional solution techniques—such as high‑order Runge‑Kutta schemes, linear‑multistep methods, or sophisticated predictor‑corrector algorithms—provide high accuracy but require complex implementation, substantial computational effort, and often involve solving implicit nonlinear systems at each step.

The authors propose to approximate the solution simply by expanding n(t) and each C_i(t) to first order:

 n(t+Δt) ≈ n(t) + Δt·(dn/dt)_t,

 C_i(t+Δt) ≈ C_i(t) + Δt·(dC_i/dt)_t,

where the time derivatives are directly obtained from the point‑kinetics equations themselves. No additional matrix inversions or iterative solvers are needed; the update is explicit and scales linearly with the number of precursor groups. The method therefore offers a transparent algorithm that can be coded in a few lines of any programming language.

To assess accuracy and robustness, the authors test the scheme against four representative reactivity histories: (1) a step insertion (ρ = 0.003), (2) a linear ramp (ρ = 0.001 t), (3) a sinusoidal modulation (ρ = 0.0005 sin ωt), and (4) a zig‑zag (alternating ±0.001) pattern. For each case, they examine several time‑step sizes (Δt = 0.001 s, 0.005 s, 0.01 s) and compare results with those obtained from the widely used Prudent‑Henri method, a high‑order Runge‑Kutta implementation, and a fixed‑point iteration technique.

The step‑insertion test shows that the neutron‑density peak, the subsequent decay rate, and the precursor dynamics are reproduced with an average absolute error below 0.1 % relative to the benchmark solutions. The ramp case demonstrates that the method captures the smooth increase in neutron population without any artificial overshoot, matching the high‑order Runge‑Kutta results to within 0.08 % error. In the sinusoidal scenario, frequency‑domain analysis reveals a phase lag of less than 0.2° and an amplitude deviation under 0.15 % across the entire simulation window. The zig‑zag test, which stresses the algorithm with rapid sign changes in reactivity, still yields accurate tracking of both neutron and precursor oscillations, confirming the method’s stability under abrupt reactivity transients.

Error analysis indicates that the truncation error grows proportionally with Δt, as expected for a first‑order scheme. Nevertheless, for Δt ≤ 0.005 s the method maintains high fidelity, while larger steps (Δt ≈ 0.01 s) begin to exhibit noticeable drift. The authors also explore extending the Taylor expansion to second order; this modest increase in computational work reduces the error by roughly an order of magnitude, making the approach competitive with traditional second‑order schemes even for problems with large delayed‑neutron fractions (β) or strong feedback effects.

From a performance standpoint, the explicit first‑order Taylor algorithm consumes about 30 % less CPU time than a comparable fourth‑order Runge‑Kutta implementation for the same time step and simulation length, and its memory footprint scales linearly with the number of precursor groups (O(N)). Consequently, the method is well suited for real‑time reactor control applications, educational simulators, and rapid prototyping where simplicity and speed are paramount.

In conclusion, the paper demonstrates that a simple first‑order Taylor series expansion provides a highly accurate, stable, and computationally efficient solution to the point‑kinetics equations. The technique’s transparency makes it attractive for teaching and for integration into larger multi‑physics codes. Future work suggested by the authors includes higher‑order expansions for enhanced precision, coupling with temperature or feedback models, and application to stochastic or spatially distributed kinetics problems.