public class AdamsMoultonIntegrator extends AdamsIntegrator
Adams-Moulton methods (in fact due to Adams alone) are implicit multistep ODE solvers. This implementation is a variation of the classical one: it uses adaptive stepsize to implement error control, whereas classical implementations are fixed step size. The value of state vector at step n+1 is a simple combination of the value at step n and of the derivatives at steps n+1, n, n-1 ... Since y'n+1 is needed to compute yn+1, another method must be used to compute a first estimate of yn+1, then compute y'n+1, then compute a final estimate of yn+1 using the following formulas. Depending on the number k of previous steps one wants to use for computing the next value, different formulas are available for the final estimate:
A k-steps Adams-Moulton method is of order k+1.
We define scaled derivatives si(n) at step n as:
s1(n) = h y'n for first derivative s2(n) = h2/2 y''n for second derivative s3(n) = h3/6 y'''n for third derivative ... sk(n) = hk/k! y(k)n for kth derivative
The definitions above use the classical representation with several previous first derivatives. Lets define
qn = [ s1(n-1) s1(n-2) ... s1(n-(k-1)) ]T(we omit the k index in the notation for clarity). With these definitions, Adams-Moulton methods can be written:
Instead of using the classical representation with first derivatives only (yn, s1(n+1) and qn+1), our implementation uses the Nordsieck vector with higher degrees scaled derivatives all taken at the same step (yn, s1(n) and rn) where rn is defined as:
rn = [ s2(n), s3(n) ... sk(n) ]T(here again we omit the k index in the notation for clarity)
Taylor series formulas show that for any index offset i, s1(n-i) can be computed from s1(n), s2(n) ... sk(n), the formula being exact for degree k polynomials.
s1(n-i) = s1(n) + ∑j j (-i)j-1 sj(n)The previous formula can be used with several values for i to compute the transform between classical representation and Nordsieck vector. The transform between rn and qn resulting from the Taylor series formulas above is:
qn = s1(n) u + P rnwhere u is the [ 1 1 ... 1 ]T vector and P is the (k-1)×(k-1) matrix built with the j (-i)j-1 terms:
[ -2 3 -4 5 ... ]
[ -4 12 -32 80 ... ]
P = [ -6 27 -108 405 ... ]
[ -8 48 -256 1280 ... ]
[ ... ]
Using the Nordsieck vector has several advantages:
The predicted Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows:
[ 0 0 ... 0 0 | 0 ]
[ ---------------+---]
[ 1 0 ... 0 0 | 0 ]
A = [ 0 1 ... 0 0 | 0 ]
[ ... | 0 ]
[ 0 0 ... 1 0 | 0 ]
[ 0 0 ... 0 1 | 0 ]
From this predicted vector, the corrected vector is computed as follows:
The P-1u vector and the P-1 A P matrix do not depend on the state, they only depend on k and therefore are precomputed once for all.
MultistepIntegrator.NordsieckTransformernordsieck, scaledmainSetDimension, scalAbsoluteTolerance, scalRelativeTolerance, vecAbsoluteTolerance, vecRelativeToleranceisLastStep, resetOccurred, stepHandlers, stepSize, stepStart| Constructor and Description |
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AdamsMoultonIntegrator(int nSteps,
double minStep,
double maxStep,
double[] vecAbsoluteTolerance,
double[] vecRelativeTolerance)
Build an Adams-Moulton integrator with the given order and error control parameters.
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AdamsMoultonIntegrator(int nSteps,
double minStep,
double maxStep,
double scalAbsoluteTolerance,
double scalRelativeTolerance)
Build an Adams-Moulton integrator with the given order and error control parameters.
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| Modifier and Type | Method and Description |
|---|---|
void |
integrate(ExpandableStatefulODE equations,
double t)
Integrate a set of differential equations up to the given time.
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initializeHighOrderDerivatives, updateHighOrderDerivativesPhase1, updateHighOrderDerivativesPhase2computeStepGrowShrinkFactor, getMaxGrowth, getMinReduction, getSafety, getStarterIntegrator, setMaxGrowth, setMinReduction, setSafety, setStarterIntegrator, startfilterStep, getCurrentStepStart, getMaxStep, getMinStep, initializeStep, resetInternalState, sanityChecks, setInitialStepSize, setStepSizeControl, setStepSizeControlacceptStep, addEventHandler, addEventHandler, addStepHandler, clearEventHandlers, clearStepHandlers, computeDerivatives, getCurrentSignedStepsize, getEvaluations, getEventHandlers, getMaxEvaluations, getName, getStepHandlers, initIntegration, integrate, setEquations, setMaxEvaluations, setStateInitializedpublic AdamsMoultonIntegrator(int nSteps,
double minStep,
double maxStep,
double scalAbsoluteTolerance,
double scalRelativeTolerance)
nSteps - number of steps of the method excluding the one being computedminStep - 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 errorNumberIsTooSmallException - if order is 1 or lesspublic AdamsMoultonIntegrator(int nSteps,
double minStep,
double maxStep,
double[] vecAbsoluteTolerance,
double[] vecRelativeTolerance)
nSteps - number of steps of the method excluding the one being computedminStep - 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 errorIllegalArgumentException - if order is 1 or lesspublic void integrate(ExpandableStatefulODE equations, double t)
This method solves an Initial Value Problem (IVP).
The set of differential equations is composed of a main set, which can be extended by some sets of secondary equations. The set of equations must be already set up with initial time and partial states. At integration completion, the final time and partial states will be available in the same object.
Since this method stores some internal state variables made available in its public interface during integration
(AbstractIntegrator.getCurrentSignedStepsize()), it is not thread-safe.
integrate in class AdamsIntegratorequations - complete set of differential equations to integratet - target time for the integration
(can be set to a value smaller than t0 for backward integration)Copyright © 2017 CNES. All rights reserved.