Generalized Runge-Kutta methods
A multiple-nodes Runge-Kutta method with q stages and s derivatives can be described by the equations
y n+1 =y n +S r=1 s hr / r! S j=1 q bj (r) kj (r) ,
k (r) i= Dyr f(t n +hci (r) ,y n + S d=1 s hd / d! S j=1 q a ij (d) kj (d) ), 1<= i<= q, 1<= r<= s.
Some of the collocation methods are equivalent with these formulae.
A particular case is that of the methods based on Taylor series (for q=1).
Methods based on Gauss and Lobatto quadratures have been developed for s=3 and q<= 4 which are A(a)-stable.
The B-process or Rosenbrock 's method can be obtained applying one step of a Newton iterative process for solving the equations associated with a DIRK method:
y n+1 =yn+hS i=1 qbiki
ki=f(yn+hS j=1 i-1 a ij kj)+hd i J(yn+hS j=1 i-1 d ij kj)ki, i=1, ... ,q,
where J=d f /d y. A such method which is L-stable has at most q order.
The semi-implicit Runge-Kutta method or Rosenbrock-Wanner method (ROW) has been described as the iterative process
y n+1 =yn+hS i=1 qbiki,
Wki=f(tn+aih,yn+S j=1 i-1 a ij kj)+
hJ(tn,yn)S j=1 i-1 d ij kj+dihJ(tn,yn), i=1, ... ,q,
where W=I-dhJ(tn,yn).
A W-method is described by the following equations:
y n+1 =yn+hS i=1 qbiki
(I-hd ii A)ki=f(tn+hS j=1 i-1 a ij ,yn+hS j=1 i-1 a ij kj)+hAS j=1 i-1 d ij kj, i=1 ... ,q.
The maximum order of a W-method with q-stages is q+1.
The multistep Runge-Kutta method class includes the hybrid methods and the multistage-multistep methods:
y n+k =S j=1 kajy n+j-1 +hS j=1 qbjf(tn+cjh,Yj (n) ),
Yi (n) =S j=0 k a'ij y n+j-1 +hS j=1 q b' ij f(tn+cjh,Yj (n) ), i=1, ... ,q.
For example, the parametrized implicit Runge-Kutta methods (PIRK) or Bokhoven 's methods (IEQ)
are given by the iterative process
y n =y n-1 +h S i=1 q bi ki ,
ki =f(tn +ci h,(1-ti)y n-1 +tiy n +h S j=1 q a ij kj)
where t1=0, ci =ti+ S j=1 q a ij , 0<= ci <= 1 , i=1,...,q.
Some A-stable examples of such methods have been constructed.