Class DormandPrince853Integrator
- All Implemented Interfaces:
FirstOrderIntegrator
,ODEIntegrator
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).
- Since:
- 1.2
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Field Summary
FieldsModifier and TypeFieldDescriptionprivate static final double
First error weights array, element 1.private static final double
First error weights array, element 6.private static final double
First error weights array, element 7.private static final double
First error weights array, element 8.private static final double
First error weights array, element 9.private static final double
First error weights array, element 10.private static final double
First error weights array, element 11.private static final double
First error weights array, element 12.private static final double
Second error weights array, element 1.private static final double
Second error weights array, element 6.private static final double
Second error weights array, element 7.private static final double
Second error weights array, element 8.private static final double
Second error weights array, element 9.private static final double
Second error weights array, element 10.private static final double
Second error weights array, element 11.private static final double
Second error weights array, element 12.private static final String
Integrator method name.private static final double[][]
Internal weights Butcher array.private static final double[]
Propagation weights Butcher array.private static final double[]
Time steps Butcher array.Fields inherited from class org.apache.commons.math3.ode.nonstiff.AdaptiveStepsizeIntegrator
mainSetDimension, scalAbsoluteTolerance, scalRelativeTolerance, vecAbsoluteTolerance, vecRelativeTolerance
Fields inherited from class org.apache.commons.math3.ode.AbstractIntegrator
isLastStep, resetOccurred, stepHandlers, stepSize, stepStart
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Constructor Summary
ConstructorsConstructorDescriptionDormandPrince853Integrator
(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 TypeMethodDescriptionprotected double
estimateError
(double[][] yDotK, double[] y0, double[] y1, double h) Compute the error ratio.int
getOrder()
Get the order of the method.Methods inherited from class org.apache.commons.math3.ode.nonstiff.EmbeddedRungeKuttaIntegrator
getMaxGrowth, getMinReduction, getSafety, integrate, setMaxGrowth, setMinReduction, setSafety
Methods inherited from class org.apache.commons.math3.ode.nonstiff.AdaptiveStepsizeIntegrator
filterStep, getCurrentStepStart, getMaxStep, getMinStep, initializeStep, resetInternalState, sanityChecks, setInitialStepSize, setStepSizeControl, setStepSizeControl
Methods inherited from class org.apache.commons.math3.ode.AbstractIntegrator
acceptStep, addEventHandler, addEventHandler, addStepHandler, clearEventHandlers, clearStepHandlers, computeDerivatives, getCounter, getCurrentSignedStepsize, getEvaluations, getEvaluationsCounter, getEventHandlers, getExpandable, getMaxEvaluations, getName, getStepHandlers, initIntegration, integrate, setEquations, setMaxEvaluations, setStateInitialized
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Field Details
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METHOD_NAME
Integrator method name.- See Also:
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STATIC_C
private static final double[] STATIC_CTime steps Butcher array. -
STATIC_A
private static final double[][] STATIC_AInternal weights Butcher array. -
STATIC_B
private static final double[] STATIC_BPropagation weights Butcher array. -
E1_01
private static final double E1_01First error weights array, element 1.- See Also:
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E1_06
private static final double E1_06First error weights array, element 6.- See Also:
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E1_07
private static final double E1_07First error weights array, element 7.- See Also:
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E1_08
private static final double E1_08First error weights array, element 8.- See Also:
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E1_09
private static final double E1_09First error weights array, element 9.- See Also:
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E1_10
private static final double E1_10First error weights array, element 10.- See Also:
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E1_11
private static final double E1_11First error weights array, element 11.- See Also:
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E1_12
private static final double E1_12First error weights array, element 12.- See Also:
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E2_01
private static final double E2_01Second error weights array, element 1.- See Also:
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E2_06
private static final double E2_06Second error weights array, element 6.- See Also:
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E2_07
private static final double E2_07Second error weights array, element 7.- See Also:
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E2_08
private static final double E2_08Second error weights array, element 8.- See Also:
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E2_09
private static final double E2_09Second error weights array, element 9.- See Also:
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E2_10
private static final double E2_10Second error weights array, element 10.- See Also:
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E2_11
private static final double E2_11Second error weights array, element 11.- See Also:
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E2_12
private static final double E2_12Second error weights array, element 12.- See Also:
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Constructor Details
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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 (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than thismaxStep
- maximal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than thisscalAbsoluteTolerance
- allowed absolute errorscalRelativeTolerance
- allowed relative error
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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 (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than thismaxStep
- maximal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than thisvecAbsoluteTolerance
- allowed absolute errorvecRelativeTolerance
- allowed relative error
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Method Details
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getOrder
public int getOrder()Get the order of the method.- Specified by:
getOrder
in classEmbeddedRungeKuttaIntegrator
- Returns:
- order of the method
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estimateError
protected double estimateError(double[][] yDotK, double[] y0, double[] y1, double h) Compute the error ratio.- Specified by:
estimateError
in classEmbeddedRungeKuttaIntegrator
- Parameters:
yDotK
- derivatives computed during the first stagesy0
- estimate of the step at the start of the stepy1
- estimate of the step at the end of the steph
- current step- Returns:
- error ratio, greater than 1 if step should be rejected
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