org.jscience.mathematics.analysis.ode
Class DormandPrince853Integrator

java.lang.Object
  extended by org.jscience.mathematics.analysis.ode.AdaptiveStepsizeIntegrator
      extended by org.jscience.mathematics.analysis.ode.RungeKuttaFehlbergIntegrator
          extended by org.jscience.mathematics.analysis.ode.DormandPrince853Integrator
All Implemented Interfaces:
FirstOrderIntegrator, Named

public class DormandPrince853Integrator
extends RungeKuttaFehlbergIntegrator
implements Named

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

This integrator is an embedded Runge-Kutta-Fehlberg 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.jscience.mathematics.analysis.ode.AdaptiveStepsizeIntegrator
handler, scalAbsoluteTolerance, scalRelativeTolerance, switchesHandler, vecAbsoluteTolerance, vecRelativeTolerance
 
Constructor Summary
DormandPrince853Integrator(double minStep, double maxStep, double[] vecAbsoluteTolerance, double[] vecRelativeTolerance)
          Simple constructor.
DormandPrince853Integrator(double minStep, double maxStep, double scalAbsoluteTolerance, double scalRelativeTolerance)
          Simple constructor.
 
Method Summary
protected  double estimateError(double[][] yDotK, double[] y0, double[] y1, double h)
          Compute the error ratio.
 java.lang.String getName()
          Get the name of the method.
 int getOrder()
          Get the order of the method.
 
Methods inherited from class org.jscience.mathematics.analysis.ode.RungeKuttaFehlbergIntegrator
getMaxGrowth, getMinReduction, getSafety, integrate, setMaxGrowth, setMinReduction, setSafety
 
Methods inherited from class org.jscience.mathematics.analysis.ode.AdaptiveStepsizeIntegrator
addSwitchingFunction, filterStep, getMaxStep, getMinStep, getStepHandler, initializeStep, setInitialStepSize, setStepHandler
 
Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
 

Constructor Detail

DormandPrince853Integrator

public DormandPrince853Integrator(double minStep,
                                  double maxStep,
                                  double scalAbsoluteTolerance,
                                  double scalRelativeTolerance)
Simple constructor. Build an eighth order Dormand-Prince integrator with the given step bounds

Parameters:
minStep - minimal step (must be positive even for backward integration), the last step can be smaller than this
maxStep - maximal step (must be positive even for backward integration)
scalAbsoluteTolerance - allowed absolute error
scalRelativeTolerance - allowed relative error

DormandPrince853Integrator

public DormandPrince853Integrator(double minStep,
                                  double maxStep,
                                  double[] vecAbsoluteTolerance,
                                  double[] vecRelativeTolerance)
Simple constructor. Build an eighth order Dormand-Prince integrator with the given step bounds

Parameters:
minStep - minimal step (must be positive even for backward integration), the last step can be smaller than this
maxStep - maximal step (must be positive even for backward integration)
vecAbsoluteTolerance - allowed absolute error
vecRelativeTolerance - allowed relative error
Method Detail

getName

public java.lang.String getName()
Get the name of the method.

Specified by:
getName in interface FirstOrderIntegrator
Specified by:
getName in interface Named
Specified by:
getName in class RungeKuttaFehlbergIntegrator
Returns:
name of the method

getOrder

public int getOrder()
Get the order of the method.

Specified by:
getOrder in class RungeKuttaFehlbergIntegrator
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 RungeKuttaFehlbergIntegrator
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