Fehlberg

(Source code, png)

../../_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.

getName()

Accessor to the object's name.

getTransitionFunction()

Transition function accessor.

hasName()

Test if the object is named.

setName(name)

Accessor to the object's name.

setTransitionFunction(transitionFunction)

Transition function accessor.

solve(*args)

Solve ODE.
__init__(*args)
getClassName()

Accessor to the object’s name.

Returns:
class_namestr

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

getName()

Accessor to the object’s name.

Returns:
namestr

The name of the object.

getTransitionFunction()

Transition function accessor.

Returns:
transitionFunctionFieldFunction

Transition function.

hasName()

Test if the object is named.

Returns:
hasNamebool

True if the name is not empty.

setName(name)

Accessor to the object’s name.

Parameters:
namestr

The name of the object.

setTransitionFunction(transitionFunction)

Transition function accessor.

Parameters:
transitionFunctionFieldFunction

Transition function.

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.