See: Description
| Interface | Description |
|---|---|
| FirstOrderDifferentialEquations |
This interface represents a first order differential equations set.
|
| FirstOrderIntegrator |
This interface represents a first order integrator for
differential equations.
|
| MainStateJacobianProvider |
Interface expanding
first order
differential equations in order to compute exactly the main state jacobian
matrix for partial derivatives equations. |
| MultistepIntegrator.NordsieckTransformer |
Transformer used to convert the first step to Nordsieck representation.
|
| ODEIntegrator |
This interface defines the common parts shared by integrators
for first and second order differential equations.
|
| Parameterizable |
This interface enables to process any parameterizable object.
|
| ParameterizedODE |
Interface to compute by finite difference Jacobian matrix for some parameter
when computing
partial derivatives equations. |
| ParameterJacobianProvider |
Interface to compute exactly Jacobian matrix for some parameter
when computing
partial derivatives equations. |
| SecondaryEquations |
This interface allows users to add secondary differential equations to a primary
set of differential equations.
|
| SecondOrderDifferentialEquations |
This interface represents a second order differential equations set.
|
| SecondOrderIntegrator |
This interface represents a second order integrator for
differential equations.
|
| Class | Description |
|---|---|
| AbstractIntegrator |
Base class managing common boilerplate for all integrators.
|
| AbstractParameterizable |
This abstract class provides boilerplate parameters list.
|
| ContinuousOutputModel |
This class stores all information provided by an ODE integrator
during the integration process and build a continuous model of the
solution from this.
|
| EquationsMapper |
Class mapping the part of a complete state or derivative that pertains
to a specific differential equation.
|
| ExpandableStatefulODE |
This class represents a combined set of first order differential equations,
with at least a primary set of equations expandable by some sets of secondary
equations.
|
| FirstOrderConverter |
This class converts second order differential equations to first
order ones.
|
| JacobianMatrices |
This class defines a set of
secondary equations to
compute the Jacobian matrices with respect to the initial state vector and, if
any, to some parameters of the primary ODE set. |
| MultistepIntegrator |
This class is the base class for multistep integrators for Ordinary
Differential Equations.
|
| Exception | Description |
|---|---|
| JacobianMatrices.MismatchedEquations |
Special exception for equations mismatch.
|
| UnknownParameterException |
Exception to be thrown when a parameter is unknown.
|
This package provides classes to solve Ordinary Differential Equations problems.
This package solves Initial Value Problems of the form
y'=f(t,y) with t0 and
y(t0)=y0 known. The provided
integrators compute an estimate of y(t) from
t=t0 to t=t1.
It is also possible to get thederivatives with respect to the initial state
dy(t)/dy(t0) or the derivatives with
respect to some ODE parameters dy(t)/dp.
All integrators provide dense output. This means that besides
computing the state vector at discrete times, they also provide a
cheap mean to get the state between the time steps. They do so through
classes extending the StepInterpolator
abstract class, which are made available to the user at the end of
each step.
All integrators handle multiple discrete events detection based on switching functions. This means that the integrator can be driven by user specified discrete events. The steps are shortened as needed to ensure the events occur at step boundaries (even if the integrator is a fixed-step integrator). When the events are triggered, integration can be stopped (this is called a G-stop facility), the state vector can be changed, or integration can simply go on. The latter case is useful to handle discontinuities in the differential equations gracefully and get accurate dense output even close to the discontinuity.
The user should describe his problem in his own classes
(UserProblem in the diagram below) which should implement
the FirstOrderDifferentialEquations interface. Then he should pass it to
the integrator he prefers among all the classes that implement the
FirstOrderIntegrator interface.
The solution of the integration problem is provided by two means. The
first one is aimed towards simple use: the state vector at the end of
the integration process is copied in the y array of the
FirstOrderIntegrator.integrate method. The second one should be used
when more in-depth information is needed throughout the integration
process. The user can register an object implementing the StepHandler interface or a
StepNormalizer
object wrapping a user-specified object implementing the FixedStepHandler
interface into the integrator before calling the FirstOrderIntegrator.integrate method. The user object will be called
appropriately during the integration process, allowing the user to
process intermediate results. The default step handler does nothing.
ContinuousOutputModel is a special-purpose step handler that is able
to store all steps and to provide transparent access to any
intermediate result once the integration is over. An important feature
of this class is that it implements the Serializable
interface. This means that a complete continuous model of the
integrated function throughout the integration range can be serialized
and reused later (if stored into a persistent medium like a filesystem
or a database) or elsewhere (if sent to another application). Only the
result of the integration is stored, there is no reference to the
integrated problem by itself.
Other default implementations of the StepHandler interface are
available for general needs (DummyStepHandler, StepNormalizer) and custom
implementations can be developed for specific needs. As an example,
if an application is to be completely driven by the integration
process, then most of the application code will be run inside a step
handler specific to this application.
Some integrators (the simple ones) use fixed steps that are set at
creation time. The more efficient integrators use variable steps that
are handled internally in order to control the integration error with
respect to a specified accuracy (these integrators extend the AdaptiveStepsizeIntegrator abstract class). In this case, the step
handler which is called after each successful step shows up the
variable stepsize. The StepNormalizer class can
be used to convert the variable stepsize into a fixed stepsize that
can be handled by classes implementing the FixedStepHandler
interface. Adaptive stepsize integrators can automatically compute the
initial stepsize by themselves, however the user can specify it if he
prefers to retain full control over the integration or if the
automatic guess is wrong.
| Fixed Step Integrators | |
| Name | Order |
Euler | 1 |
Midpoint | 2 |
Classical Runge-Kutta
| 4 |
Gill | 4 |
3/8 | 4 |
| Adaptive Stepsize Integrators | ||
| Name | Integration Order | Error Estimation Order |
Higham and Hall
| 5 | 4 |
Dormand-Prince 5(4)
| 5 | 4 |
Dormand-Prince 8(5,3)
| 8 | 5 and 3 |
Gragg-Bulirsch-Stoer
| variable (up to 18 by default) | variable |
Adams-Bashforth
| variable | variable |
Adams-Moulton
| variable | variable |
In the table above, the Adams-Bashforth and Adams-Moulton integrators appear as variable-step ones. This is an experimental extension
to the classical algorithms using the Nordsieck vector representation.
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