Fehlberg

(Source code, png, hires.png, pdf)

../../_images/openturns-Fehlberg-1.png
class Fehlberg(*args)

Adaptive order Fehlberg method.

Parameters
transitionFunctionFunction

The function defining the flow of the ordinary differential equation. Must have one parameter.

localPrecisionfloat

The expected absolute error on one step.

orderint, order\in\{0,1,2,3,4\}

The order of the method, ie the exponent p in the estimate of the local error for a step of size h written as \cO(h^p).

See also

ODESolver

Notes

The Fehlberg method of order p\in\Nset is a one-step explicit method made of two embedded Runge Kutta methods of order p and p+1. More precisely, such a method approximate the solution of:

\vect{y}'(t)=f\left(t,\vect{y}(t)\right)\quad\mbox{with}\quad \vect{y}(t_0)=\vect{y}_0

at a given set of locations t_0,\dots,t_N by first building an approximation over an adapted grid \tau_0=t_0,\dots,\tau_M=t_N with a number of points M not necessarily equal to the number of locations N and internal nodes not necessarily part of the set of locations. Then, the solution \vect{y} is approximated by a smooth piecewise polynomial function using PiecewiseHermiteEvaluation, which is evaluated over the set of locations.

The method proceeds as follows. Knowing the solution at location \vect{y}_i=\vect{y}(\tau_i) and a current time step h_i, two approximations \hat{\vect{y}}_{i+1} and \bar{\vect{y}}_{i+1} of \vect{y}_{i+1}=\vect{y}(\tau_i+h_i)=\vect{y}(\tau_{i+1}) are built, such that:

\hat{\vect{y}}_{i+1}=\vect{y}_i+h_i\vect{\Phi}_{\mathrm{I}}\left(\tau_i, \vect{y}_i, h_i\right) \\ \bar{\vect{y}}_{i+1}=\vect{y}_i+h_i\vect{\Phi}_{\mathrm{II}}\left(\tau_i, \vect{y}_i, h_i\right)

where we assume that:

\left|\vect{\Phi}_{\mathrm{I}}\left(\tau_i,\vect{y}_i,h_i\right)-(\vect{y}_{i+1}-\vect{y}_i)/h_i\right|=\cO\left(h_i^p\right)\\ \left|\vect{\Phi}_{\mathrm{II}}\left(\tau_i, \vect{y}_i,h_i\right)-(\vect{y}_{i+1}-\vect{y}_i)/h_i\right|=\cO\left(h_i^{p+1}\right)

The evolution operators \vect{\Phi}_{\mathrm{I}} and \vect{\Phi}_{\mathrm{II}} are constructed as follows:

\vect{\Phi}_{\mathrm{I}}\left(\tau, \vect{y}_i, h_i\right)=\sum_{k=0}^p c_kf_k\left(\tau,\vect{y}_i; h_i\right)\\ \vect{\Phi}_{\mathrm{II}}\left(\tau, \vect{y}_i, h_i\right)=\sum_{k=0}^{p+1} \hat{c}_kf_k\left(\tau,\vect{y}_i; h_i\right)

with f_k=f_k\left(\tau,\vect{y}_i,h_i\right)=f\left(\tau+\alpha_kh_i,\vect{y}_i+h_i\sum_{\ell=0}^{k-1}\beta_{k\ell}f_{\ell}\right). The most desirable property of these methods is their embedded nature: the high-order approximation reuses all the evaluations of f needed by the low-order approximation. The coefficients c_k, \hat{c}_k, \alpha_k and \beta_{k\ell} fully specify the method.

For p=0 we have:

k

\alpha_k

\beta_{k0}

c_k

\hat{c}_k

0

0

0

1

1/2

1

1

1

1/2

For p=1 we have:

k

\alpha_k

\beta_{k0}

\beta_{k1}

c_k

\hat{c}_k

0

0

0

1/256

1/512

1

1/2

1/2

255/256

255/256

2

1

1/256

255/256

1/512

For p=2 we have:

k

\alpha_k

\beta_{k0}

\beta_{k1}

\beta_{k2}

c_k

\hat{c}_k

0

0

0

214/891

533/2106

1

1/4

1/4

1/33

0

2

27/40

-189/800

214/891

650/891

800/1053

3

1

729/800

1/35

650/891

-1/78

For p>2 the coefficients can be found eg in the C++ source code. For additional theory on these methods see [stoer1993], chapter 7.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['t', 'y0', 'y1'], ['t - y0', 'y1 + t^2'])
>>> phi = ot.ParametricFunction(f, [0], [0.0])
>>> solver = ot.Fehlberg(phi)
>>> Y0 = [1.0, -1.0]
>>> nt = 100
>>> timeGrid = [(i**2.0) / (nt - 1.0)**2.0 for i in range(nt)]
>>> result = solver.solve(Y0, timeGrid)

Methods

getClassName()

Accessor to the object’s name.

getId()

Accessor to the object’s id.

getName()

Accessor to the object’s name.

getShadowedId()

Accessor to the object’s shadowed id.

getTransitionFunction()

Transition function accessor.

getVisibility()

Accessor to the object’s visibility state.

hasName()

Test if the object is named.

hasVisibleName()

Test if the object has a distinguishable name.

setName(name)

Accessor to the object’s name.

setShadowedId(id)

Accessor to the object’s shadowed id.

setTransitionFunction(transitionFunction)

Transition function accessor.

setVisibility(visible)

Accessor to the object’s visibility state.

solve(*args)

Solve ODE.
__init__(*args)

Initialize self. See help(type(self)) for accurate signature.

getClassName()

Accessor to the object’s name.

Returns
class_namestr

The object class name (object.__class__.__name__).

getId()

Accessor to the object’s id.

Returns
idint

Internal unique identifier.

getName()

Accessor to the object’s name.

Returns
namestr

The name of the object.

getShadowedId()

Accessor to the object’s shadowed id.

Returns
idint

Internal unique identifier.

getTransitionFunction()

Transition function accessor.

Returns
transitionFunctionFieldFunction

Transition function.

getVisibility()

Accessor to the object’s visibility state.

Returns
visiblebool

Visibility flag.

hasName()

Test if the object is named.

Returns
hasNamebool

True if the name is not empty.

hasVisibleName()

Test if the object has a distinguishable name.

Returns
hasVisibleNamebool

True if the name is not empty and not the default one.

setName(name)

Accessor to the object’s name.

Parameters
namestr

The name of the object.

setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters
idint

Internal unique identifier.

setTransitionFunction(transitionFunction)

Transition function accessor.

Parameters
transitionFunctionFieldFunction

Transition function.

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters
visiblebool

Visibility flag.

solve(*args)

Solve ODE.

Parameters
initialStatesequence of float

Initial value of the equation

timeGridsequence of float or Mesh of dimension 1

Time stamps, ie values of t at which the solution is computed.

Returns
valuesSample

The solution of the equation at grid points.