Examples of org.apache.commons.math.ode.nonstiff.AdamsBashforthIntegrator

• org.apache.commons.math.ode.nonstiff.AdamsBashforthIntegrator
  This class implements explicit Adams-Bashforth integrators for Ordinary Differential Equations.

  Adams-Bashforth methods (in fact due to Adams alone) are explicit 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, n-1, n-2 ... Depending on the number k of previous steps one wants to use for computing the next value, different formulas are available:

  • k = 1: y_n+1 = y_n + h y'_n
  • k = 2: y_n+1 = y_n + h (3y'_n-y'_n-1)/2
  • k = 3: y_n+1 = y_n + h (23y'_n-16y'_n-1+5y'_n-2)/12
  • k = 4: y_n+1 = y_n + h (55y'_n-59y'_n-1+37y'_n-2-9y'_n-3)/24
  • ...

  A k-steps Adams-Bashforth method is of order k.

  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-Bashforth methods can be written:

  • k = 1: y_n+1 = y_n + s₁(n)
  • k = 2: y_n+1 = y_n + 3/2 s₁(n) + [ -1/2 ] q_n
  • k = 3: y_n+1 = y_n + 23/12 s₁(n) + [ -16/12 5/12 ] q_n
  • k = 4: y_n+1 = y_n + 55/24 s₁(n) + [ -59/24 37/24 -9/24 ] q_n
  • ...

  Instead of using the classical representation with first derivatives only (y_n, s₁(n) and q_n), 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 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 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 ] 

  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.

  @version $Revision: 1073158 $ $Date: 2011-02-21 22:46:52 +0100 (lun. 21 févr. 2011) $ @since 2.0

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