Robertson

class otp.robertson.RobertsonProblem

Bases: otp.Problem

A simple, stiff chemical reaction.

The Robertson problem [Rob66] is a simple, stiff chemical reaction that models the concentrations of chemical species $\ce{A}$, $\ce{B}$, and $\ce{C}$ involved in the reactions

\[\begin{split} \ce{ A &->[$k_1$] B \\ B + B &->[$k_2$] C + B \\ B + C &->[$k_3$] A + C } \end{split}\]

These correspond to the ODE system,

\[\begin{split} y_1' &= -k_1 y_1 + k_3 y_2 y_3, \\ y_2' &= k_1 y_1 - k_2 y_2^2 - k_3 y_2 y_3, \\ y_3' &= k_2 y_2^2. \end{split}\]

The reaction rates $k_1$, $k_2$, and $k_3$ often range from slow to very fast which makes the problem challenging. This has made it a popular test for implicit time-stepping schemes.

Notes

Type	ODE
Number of Variables	3
Stiff	typically, depending on $k_1$, $k_2$, and $k_3$

Example

>>> problem = otp.robertson.presets.Canonical;
>>> problem.Parameters.K1 = 100;
>>> problem.Parameters.K2 = 1000;
>>> problem.Parameters.K3 = 1000;
>>> sol = problem.solve();
>>> problem.plot(sol)

Constructor Summary

RobertsonProblem(timeSpan, y0, parameters)

Create a Robertson problem object.

Parameters:

timeSpan (numeric(1, 2)) – The start and final time.
y0 (numeric(3, 1)) – The initial conditions.
parameters (RobertsonParameters) – The parameters.

Parameters

class otp.robertson.RobertsonParameters

Bases: otp.Parameters

Parameters for the Robertson problem.

Constructor Summary

RobertsonParameters(varargin)

Create a Robertson parameters object.

Parameters:: varargin – A variable number of name-value pairs. A name can be any property of this class, and the subsequent value initializes that property.

Property Summary

K1: The reaction rate $k_1$.

K2: The reaction rate $k_2$.

K3: The reaction rate $k_3$.

Presets

class otp.robertson.presets.Canonical

Bases: otp.robertson.RobertsonProblem

The original configuration [Rob66] with $t ∈ [0, 40]$, $y_0 = [1, 0, 0]^T$, and parameters

\[\begin{split} k_1 &= 4 \times 10^{-2}, \\ k_2 &= 3 \times 10^7, \\ k_3 &= 10^4. \end{split}\]

Constructor Summary

Canonical(varargin)

Create the canonical Robertson problem object.

Parameters:

varargin –

A variable number of name-value pairs. The accepted names are

K1 – Value of $k_1$.
K2 – Value of $k_2$.
K3 – Value of $k_3$.

class otp.robertson.presets.ROBER

Bases: otp.robertson.RobertsonProblem

The Robertson configuration from [HW96] (p. 144). This differs from Canonical only in the time span which is extended to $[0, 10^{11}]$. This presents a challenge for numerical integrators to preserve solution positivity.

Constructor Summary

ROBER(): Create the ROBER Robertson problem object.