Solvers¶

SCIANTIX relies on a set of numerical solvers capable of handling Ordinary Differential Equations (ODEs), Partial Differential Equations (PDEs), and non-linear equations. These solvers are implemented in the Solver class.

ODE Solvers¶

The code includes specific solvers for first-order ODEs of the form:

\[\frac{dy}{dt} = S - L y\]

where \(S\) is a source term and \(L\) is a loss (decay) rate.

Integrator¶

Solves the simple integration \(y' = S\) (i.e., \(L=0\)) using a forward Euler scheme:

\[y^{n+1} = y^n + S \Delta t\]

Usage: Used for cumulative quantities like burnup, gas production, and effective burnup.

Decay¶

Solves the decay equation \(y' = S - L y\) using an analytical solution over the time step (assuming constant coefficients):

\[y^{n+1} = \frac{y^n + S \Delta t}{1 + L \Delta t}\]

Usage: Used for radioactive decay, grain boundary micro-cracking retention, and intragranular bubble evolution.

Limited Growth¶

Solves a logistic-type growth equation \(y' = k/y + S\).

Binary Interaction¶

Solves equations of the form \(y' = -k y^2\) representing binary interactions (e.g., bubble coalescence).

PDE Solvers¶

SCIANTIX solves diffusion-reaction equations using a Spectral Method.

Spectral Diffusion¶

Solves the diffusion equation in spherical symmetry:

\[\frac{\partial C}{\partial t} = D \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial C}{\partial r} \right) - \beta C + S\]

The spatial dependence is handled by projecting the concentration \(C(r,t)\) onto a basis of eigenfunctions of the Laplacian operator (diffusion modes). This transforms the PDE into a system of decoupled ODEs for the mode amplitudes, which are then solved analytically over the time step.

Coupled Diffusion¶

SpectralDiffusion2equations: Solves a system of 2 coupled diffusion equations.
SpectralDiffusion3equations: Solves a system of 3 coupled diffusion equations.

Non-linear Solvers¶

Newton-Blackburn¶

Solves the Blackburn thermochemical model equation using the Newton-Raphson method to find the equilibrium stoichiometry deviation \(x\) (in \(UO_{2+x}\)) given temperature and oxygen partial pressure.

Newton-Langmuir¶

Solves the Langmuir-based stoichiometry deviation model using the Newton-Raphson method.

Quartic Equation¶

Solves a quartic equation \(ax^4 + bx^3 + cx^2 + dx + e = 0\) using Newton’s method. Used in the grain growth model.

Linear Algebra¶

The Solver class provides utility methods for linear algebra, such as:

Laplace: Solves systems of linear equations of size NxN using Cramer’s rule / Laplace expansion (optimized for N=2 and N=3).
Dot Product: Computes dot products for 1D vectors and 2D matrices.