# Fehlberg¶

class Fehlberg(*args)

Adaptive order Fehlberg method.

Parameters: transitionFunction : Function The function defining the flow of the ordinary differential equation. Must have one parameter. localPrecision : float The expected absolute error on one step. order : int, The order of the method, ie the exponent in the estimate of the local error for a step of size written as .

Notes

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

at a given set of locations by first building an approximation over an adapted grid with a number of points not necessarily equal to the number of locations and internal nodes not necessarily part of the set of locations. Then, the solution 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 and a current time step , two approximations and of are built, such that:

where we assume that:

The evolution operators and are constructed as follows:

with . The most desirable property of these methods is their embedded nature: the high-order approximation reuses all the evaluations of needed by the low-order approximation. The coefficients , , and fully specify the method.

For we have:

0 0 0 1 1/2
1 1 1   1/2

For we have:

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 we have:

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 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_name : str The object class name (object.__class__.__name__).
getId()

Accessor to the object’s id.

Returns: id : int Internal unique identifier.
getName()

Accessor to the object’s name.

Returns: name : str The name of the object.
getShadowedId()

Accessor to the object’s shadowed id.

Returns: id : int Internal unique identifier.
getTransitionFunction()

Transition function accessor.

Returns: transitionFunction : FieldFunction Transition function.
getVisibility()

Accessor to the object’s visibility state.

Returns: visible : bool Visibility flag.
hasName()

Test if the object is named.

Returns: hasName : bool True if the name is not empty.
hasVisibleName()

Test if the object has a distinguishable name.

Returns: hasVisibleName : bool True if the name is not empty and not the default one.
setName(name)

Accessor to the object’s name.

Parameters: name : str The name of the object.
setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters: id : int Internal unique identifier.
setTransitionFunction(transitionFunction)

Transition function accessor.

Parameters: transitionFunction : FieldFunction Transition function.
setVisibility(visible)

Accessor to the object’s visibility state.

Parameters: visible : bool Visibility flag.
solve(*args)

Solve ODE.

Parameters: initialState : sequence of float Initial value of the equation timeGrid : sequence of float or Mesh of dimension 1 Time stamps, ie values of at which the solution is computed. values : Sample The solution of the equation at grid points.