Neutron Point Kinetics with Ramp Reactivity

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

Purpose

The goal of this section is to document how the analytical solutions for (n(t)) and (C(t)) are validated against a high-precision numerical reference obtained with a fourth–order Runge–Kutta (RK4) method.

RK4 Reference Solver

The script RK4_reference_mpmath.py integrates the NPKE system

\[\frac{dn}{dt}=\frac{\rho(t)-\beta}{\Lambda}\,n(t)+\lambda\,C(t)+q, \qquad \frac{dC}{dt}=\frac{\beta}{\Lambda}\,n(t)-\lambda\,C(t),\]

with (\rho(t)=at+b), using a sufficiently small time step and a working precision of 32 decimal digits.

The initial conditions are

\[n(0) = \frac{q\,\Lambda}{|\rho(0)|}, \qquad C(0) = \frac{\beta}{\lambda\Lambda}\,n(0),\]

corresponding to a stationary state at (t=0) for the initial reactivity value.

Error Measures

To compare the analytical and numerical solutions, absolute percentage errors (APE) are computed, for example:

\[\operatorname{APE}_n(t_k) = 100\,\frac{\bigl|n_{\text{RK4}}(t_k)-n_{\text{analytic}}(t_k)\bigr|} {\bigl|n_{\text{RK4}}(t_k)\bigr|},\]

and analogously for the precursor concentration (C(t)).

Tables similar to those reported in the manuscript can be reproduced by:

  1. Running RK4_reference_mpmath.py to generate the reference solution.
  2. Running Neutron_density_SciPyNumPy.py and C_precursor_SciPyNumPy.py for the same time grid.
  3. Computing the APE values and exporting them to a CSV or LaTeX table.

Further details on the numerical tests (ramp parameters, time grids and tolerance values) can be found in the article.