Neutron Point Kinetics with Ramp Reactivity

Analytical and numerical solutions of the NPKE with a linear reactivity ramp

Overview

The present repository documents the computational implementation of a new analytical solution of the following system of differential equations:

$$ \begin{matrix}\dfrac{dn\left(t\right)}{dt}&=&\dfrac{\rho\left(t\right)-\beta}{\Lambda}n\left(t\right)+\lambda C\left(t\right)+q,\\\dfrac{dC\left(t\right)}{dt}&=&\dfrac{\beta}{\Lambda}n\left(t\right)-\lambda C\left(t\right).\\\end{matrix}\, $$

known as the Neutron Point Kinetic Equations (NPKE). Such analytical solution corresponds to the particular case of a linear ramp reactivity, \(\rho(t)=at+b\) and was obtained using the Modified Integral Method (MIM).

Python Codes and Implementations

Python 3 implementations included in this repository:

  • Analytical neutron density \( n(t) \)
    Implementation using SciPy/NumPy libraries.
  • High-precision analytical neutron density \( n(t) \)
    Implementation using mpmath with 32-digit precision by default.
  • Delayed neutron precursor concentration \( C(t) \)
    Analytical evaluation consistent with the neutron density solvers.
  • RK4 reference solver
    A high-precision classical Runge–Kutta 4 (RK4) scheme used as benchmark.
  • Analytical benchmark solution by Zhang et al. (2008)
    Closed-form formulation for comparison and validation.
  • Analytical benchmark solution by Palma et al. (2010)
    Alternative analytical solution for cross-checking the proposed method.
  • Comparative execution-time benchmark
    A dedicated performance routine that evaluates and compares the mean execution time of all methods. The implementation makes use of time,to quantify the computational cost of each approach.

The related article

These results accompany the manuscript:

A New Analytical Solution to the Neutron Point Kinetics Equations for Linear Ramp Reactivity: \(\rho(t)=a t + b\). A Theoretical and Computational Framework

that was recently submitted to the journal Computer Physics Communications.

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Equations
Main analytical expressions and key formulae used in the article.
Codes
Summary of the Python scripts, interfaces, and numerical settings.
Validation
Comparison against the high-precision RK4 benchmark and analytical references.
About
Authorship, affiliations, and financial support acknowledgements.
⚠️ Important:
For more details of the derivation of the analytical solution, its computational implementation as well as a numerical analysis, please see the manuscript mentioned above.