Class AdamsMoultonIntegrator

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 y_n+1, another method must be used to compute a first estimate of y_n+1, then compute y'_n+1, then compute a final estimate of y_n+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: y_n+1 = y_n + h y'_n+1
• k = 2: y_n+1 = y_n + h (y'_n+1+y'_n)/2
• k = 3: y_n+1 = y_n + h (5y'_n+1+8y'_n-y'_n-1)/12
• k = 4: y_n+1 = y_n + 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 s_i(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: y_n+1 = y_n + s₁(n+1)
• k = 2: y_n+1 = y_n + 1/2 s₁(n+1) + [ 1/2 ] q_n+1
• k = 3: y_n+1 = y_n + 5/12 s₁(n+1) + [ 8/12 -1/12 ] q_n+1
• k = 4: y_n+1 = y_n + 9/24 s₁(n+1) + [ 19/24 -5/24 1/24 ] q_n+1
• ...

Instead of using the classical representation with first derivatives only (y_n, s₁(n+1) and q_n+1), our implementation uses the Nordsieck vector with higher degrees scaled derivatives all taken at the same step (y_n, s₁(n) and r_n) where r_n 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) ... s_k(n), the formula being exact for degree k polynomials.

 s1(n-i) = s1(n) + ∑j>0 (j+1) (-i)j sj+1(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+1) (-i)^j terms with i being the row number starting from 1 and j being the column number starting from 1:

        [  -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:

• Y_n+1 = y_n + s₁(n) + u^T r_n
• S₁(n+1) = h f(t_n+1, Y_n+1)
• R_n+1 = (s₁(n) - S₁(n+1)) P^-1 u + P^-1 A P r_n

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:

• y_n+1 = y_n + S₁(n+1) + [ -1 +1 -1 +1 ... ±1 ] r_n+1
• s₁(n+1) = h f(t_n+1, y_n+1)
• r_n+1 = R_n+1 + (s₁(n+1) - S₁(n+1)) P^-1 u

where the upper case Y_n+1, S₁(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.