turboflow.pysolver_view.nonlinear_system_problems module

class turboflow.pysolver_view.nonlinear_system_problems.LorentzEquations(sigma=1.0, beta=2.0, rho=3.0)

Bases: NonlinearSystemProblem

Implementation of the Lorentz System of Nonlinear Equations.

This class implements the following system of algebraic nonlinear equations:

\[\begin{split}\begin{align} \dot{x} &= \sigma(y - x) = 0\\ \dot{y} &= x(\rho - z) - y = 0\\ \dot{z} &= xy - \beta z = 0 \end{align}\end{split}\]

Where:

• \(\sigma\) is related to the Prandtl number

• \(\rho\) is related to the Rayleigh number

• \(\beta\) is a geometric factor

References

• Edward N. Lorenz. “Deterministic Nonperiodic Flow”. Journal of the Atmospheric Sciences, 20(2):130-141, 1963.

Attributes:

sigmafloat

The Prandtl number.

betafloat

The geometric factor.

rhofloat

The Rayleigh number.

Methods

evaluate_problem(vars)`:

Evaluate the Lorentz system at a given state.

gradient(x)

residual(vars)

Evaluate the system of equations for given decision variables.

Parameters:

xarray-like

Vector of decision variables.

Returns:

array_like

Vector containing the values of the system of equations for the given decision variables.