This class implements implicit Adams-Moulton integrators for Ordinary Differential Equations.
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:
- k = 1: yn+1 = yn + h y'n+1
- k = 2: yn+1 = yn + h (y'n+1+y'n)/2
- k = 3: yn+1 = yn + h (5y'n+1+8y'n-y'n-1)/12
- k = 4: yn+1 = yn + h (9y'n+1+19y'n-5y'n-1+y'n-2)/24
- ...
A k-steps Adams-Moulton method is of order k+1.
Implementation details
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:
- k = 1: yn+1 = yn + s1(n+1)
- k = 2: yn+1 = yn + 1/2 s1(n+1) + [ 1/2 ] qn+1
- k = 3: yn+1 = yn + 5/12 s1(n+1) + [ 8/12 -1/12 ] qn+1
- k = 4: yn+1 = yn + 9/24 s1(n+1) + [ 19/24 -5/24 1/24 ] qn+1
- ...
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 r
n and q
n resulting from the Taylor series formulas above is:
qn = s1(n) u + P rn
where 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:
- it greatly simplifies step interpolation as the interpolator mainly applies Taylor series formulas,
- it simplifies step changes that occur when discrete events that truncate the step are triggered,
- it allows to extend the methods in order to support adaptive stepsize.
The predicted Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows:
- Yn+1 = yn + s1(n) + uT rn
- S1(n+1) = h f(tn+1, Yn+1)
- Rn+1 = (s1(n) - S1(n+1)) P-1 u + P-1 A P rn
where A is a rows shifting matrix (the lower left part is an identity matrix):
[ 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:
- yn+1 = yn + S1(n+1) + [ -1 +1 -1 +1 ... ±1 ] rn+1
- s1(n+1) = h f(tn+1, yn+1)
- rn+1 = Rn+1 + (s1(n+1) - S1(n+1)) P-1 u
where the upper case Y
n+1, S
1(n+1) and R
n+1 represent the predicted states whereas the lower case y
n+1, s
n+1 and r
n+1 represent the corrected states.
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.
@since 2.0