Package org.hipparchus.ode.nonstiff

Class DormandPrince853Integrator

java.lang.Object

org.hipparchus.ode.AbstractIntegrator

org.hipparchus.ode.nonstiff.AdaptiveStepsizeIntegrator

org.hipparchus.ode.nonstiff.EmbeddedRungeKuttaIntegrator

org.hipparchus.ode.nonstiff.DormandPrince853Integrator

All Implemented Interfaces:

ButcherArrayProvider, ODEIntegrator

public class DormandPrince853Integrator
extends EmbeddedRungeKuttaIntegrator

This class implements the 8(5,3) Dormand-Prince integrator for Ordinary Differential Equations.

This integrator is an embedded Runge-Kutta integrator of order 8(5,3) used in local extrapolation mode (i.e. the solution is computed using the high order formula) with stepsize control (and automatic step initialization) and continuous output. This method uses 12 functions evaluations per step for integration and 4 evaluations for interpolation. However, since the first interpolation evaluation is the same as the first integration evaluation of the next step, we have included it in the integrator rather than in the interpolator and specified the method was an fsal. Hence, despite we have 13 stages here, the cost is really 12 evaluations per step even if no interpolation is done, and the overcost of interpolation is only 3 evaluations.

This method is based on an 8(6) method by Dormand and Prince (i.e. order 8 for the integration and order 6 for error estimation) modified by Hairer and Wanner to use a 5th order error estimator with 3rd order correction. This modification was introduced because the original method failed in some cases (wrong steps can be accepted when step size is too large, for example in the Brusselator problem) and also had severe difficulties when applied to problems with discontinuities. This modification is explained in the second edition of the first volume (Nonstiff Problems) of the reference book by Hairer, Norsett and Wanner: Solving Ordinary Differential Equations (Springer-Verlag, ISBN 3-540-56670-8).

Field Summary

Fields inherited from class org.hipparchus.ode.nonstiff.AdaptiveStepsizeIntegrator

mainSetDimension, scalAbsoluteTolerance, scalRelativeTolerance, vecAbsoluteTolerance, vecRelativeTolerance

Constructor Summary

Constructors

Constructor	Description
DormandPrince853Integrator(double minStep, double maxStep, double[] vecAbsoluteTolerance, double[] vecRelativeTolerance)	Simple constructor.
DormandPrince853Integrator(double minStep, double maxStep, double scalAbsoluteTolerance, double scalRelativeTolerance)	Simple constructor.

Method Summary

Modifier and Type	Method	Description
protected org.hipparchus.ode.nonstiff.DormandPrince853StateInterpolator	createInterpolator(boolean forward, double[][] yDotK, ODEStateAndDerivative globalPreviousState, ODEStateAndDerivative globalCurrentState, EquationsMapper mapper)	Create an interpolator.
protected double	estimateError(double[][] yDotK, double[] y0, double[] y1, double h)	Compute the error ratio.
double[][]	getA()	Get the internal weights from Butcher array (without the first empty row).
double[]	getB()	Get the external weights for the high order method from Butcher array.
double[]	getC()	Get the time steps from Butcher array (without the first zero).
int	getOrder()	Get the order of the method.

Methods inherited from class org.hipparchus.ode.nonstiff.EmbeddedRungeKuttaIntegrator

getMaxGrowth, getMinReduction, getSafety, integrate, setMaxGrowth, setMinReduction, setSafety

Methods inherited from class org.hipparchus.ode.nonstiff.AdaptiveStepsizeIntegrator

filterStep, getMaxStep, getMinStep, initializeStep, resetInternalState, sanityChecks, setInitialStepSize, setStepSizeControl, setStepSizeControl

Methods inherited from class org.hipparchus.ode.AbstractIntegrator

acceptStep, addEventHandler, addEventHandler, addStepHandler, clearEventHandlers, clearStepHandlers, computeDerivatives, getCurrentSignedStepsize, getCurrentStepStart, getEquations, getEvaluations, getEvaluationsCounter, getEventHandlers, getMaxEvaluations, getName, getStepHandlers, getStepSize, getStepStart, initIntegration, isLastStep, resetOccurred, setIsLastStep, setMaxEvaluations, setStateInitialized, setStepSize, setStepStart

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Methods inherited from interface org.hipparchus.ode.ODEIntegrator

integrate, integrate

Constructor Detail

DormandPrince853Integrator

public DormandPrince853Integrator(double minStep,
                                  double maxStep,
                                  double scalAbsoluteTolerance,
                                  double scalRelativeTolerance)

Simple constructor. Build a fifth order Dormand-Prince integrator with the given step bounds

Parameters:

minStep - minimal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this

maxStep - maximal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this

scalAbsoluteTolerance - allowed absolute error

scalRelativeTolerance - allowed relative error

DormandPrince853Integrator

public DormandPrince853Integrator(double minStep,
                                  double maxStep,
                                  double[] vecAbsoluteTolerance,
                                  double[] vecRelativeTolerance)

Simple constructor. Build a fifth order Dormand-Prince integrator with the given step bounds

Parameters:

minStep - minimal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this

maxStep - maximal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this

vecAbsoluteTolerance - allowed absolute error

vecRelativeTolerance - allowed relative error

Method Detail

getC

public double[] getC()

Get the time steps from Butcher array (without the first zero).

Returns:

time steps from Butcher array (without the first zero

getA

public double[][] getA()

Get the internal weights from Butcher array (without the first empty row).

Returns:

internal weights from Butcher array (without the first empty row)

getB

public double[] getB()

Get the external weights for the high order method from Butcher array.

Returns:

external weights for the high order method from Butcher array

createInterpolator

protected org.hipparchus.ode.nonstiff.DormandPrince853StateInterpolator createInterpolator(boolean forward,
                                                                                           double[][] yDotK,
                                                                                           ODEStateAndDerivative globalPreviousState,
                                                                                           ODEStateAndDerivative globalCurrentState,
                                                                                           EquationsMapper mapper)

Create an interpolator.

Specified by:

createInterpolator in class EmbeddedRungeKuttaIntegrator

Parameters:

forward - integration direction indicator

yDotK - slopes at the intermediate points

globalPreviousState - start of the global step

globalCurrentState - end of the global step

mapper - equations mapper for the all equations

Returns:

external weights for the high order method from Butcher array

getOrder

public int getOrder()

Get the order of the method.

Specified by:

getOrder in class EmbeddedRungeKuttaIntegrator

Returns:

order of the method

estimateError

protected double estimateError(double[][] yDotK,
                               double[] y0,
                               double[] y1,
                               double h)

Compute the error ratio.

Specified by:

estimateError in class EmbeddedRungeKuttaIntegrator

Parameters:

yDotK - derivatives computed during the first stages

y0 - estimate of the step at the start of the step

y1 - estimate of the step at the end of the step

h - current step

Returns:

error ratio, greater than 1 if step should be rejected