Prothero–Robinson

class otp.protherorobinson.ProtheroRobinsonProblem

Bases: otp.Problem

The prototypical test problem for analyzing order reduction due to stiffness.

The Prothero–Robinson problem [PR74] is the linear ODE

\[y' = λ (y - φ(t)) + φ'(t).\]

This simple problem is used to test for order reduction and S-stability of time-stepping schemes. With initial condition \(y(t_0) = φ(t_0)\), the exact solution is \(y(t) = φ(t)\) independent of \(λ\). Therefore, any errors introduced by the numerical scheme can be measured easily.

Notes

Type	ODE
Number of Variables	1
Stiff	typically, depending on \(λ\)

Example

>>> p = otp.protherorobinson.presets.Canonical(-10);
>>> sol = p.solve();
>>> norm(p.solveExactly(sol.x) - sol.y)

Constructor Summary

ProtheroRobinsonProblem(timeSpan, y0, parameters)

Create a Prothero–Robinson problem object.

Parameters:

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

Parameters

class otp.protherorobinson.ProtheroRobinsonParameters

Bases: otp.Parameters

Parameters for the Prothero–Robinson problem.

Constructor Summary

ProtheroRobinsonParameters(varargin)

Create a Prothero–Robinson 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

Lambda: The stiffness parameter and eigenvalue of the Jacobian \(λ\).

Phi: The function \(φ(t)\).

DPhi: The time derivative of \(φ(t)\).

Presets

class otp.protherorobinson.presets.Canonical

Bases: otp.protherorobinson.ProtheroRobinsonProblem

The Prothero–Robinson configuration from [PR74] (p. 159) based on [SLH70] (p. 272). It uses time span \(t ∈ [0, 15]\), initial condition \(y(0) = φ(0)\), and parameters

\[\begin{split} λ &= -200, \\ φ(t) &= 10 - (10 + t) e^{-t}. \end{split}\]

Constructor Summary

Canonical(varargin)

Create the canonical Prothero–Robinson problem object.

Parameters:

varargin –

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

Lambda – The stiffness parameter and eigenvalue of the Jacobian \(λ\).
Phi – The exact solution.
DPhi – The time derivative of \(φ(t)\).