In the last experiments, we implement a 2-stage implicit Runge-Kutta method
(IRK2) in the form (3.6), whose coefficients are given in the Butcher tableau
The collocation method (22) with Lagrange polynomials [w.sub.j](s) is equivalent to the N-stage implicit Runge-Kutta method
(10a) and (10b) with coefficients
The s-stage implicit Runge-Kutta method
applied to the general nonlinear DAE of the form (2) is defined by
(1) Method M546: fourth-order symmetric DIRK methods for periodic stiff problems of Franco and Gomez (see ), a symmetric diagonally implicit Runge-Kutta method
with five stages of algebraic order 4 dispersion order 6 was proposed.
Syam, "An efficient implicit Runge-Kutta method
for second order systems," Applied Mathematics and Computation, vol.
(iv) ESDIRK: explicit singly diagonally implicit Runge-Kutta method
However, the implicit Runge-Kutta method
is not of this form, and for this reason we treat it separately in Section 3.
Bt Ismail and Suleiman  studied the P-Stability and Q-Stability of singly diagonally implicit Runge-Kutta method
for delay differential equations.
* The method RADAU5, which is an implementation of a fifth-order implicit Runge-Kutta method
of RADAU IIA type, with stages and automatic stepsize control.
Diagonally implicit Runge-Kutta methods
for stiff ODEs.
Tang, "Stability analysis of continuous implicit Runge-Kutta methods
for Volterra integro-differential systems with unbounded delays," Applied Numerical Mathematics, vol.
Hofmann, "Steady-state algorithms for nonlinear time-periodic magnetic diffusion problems using diagonally implicit Runge-Kutta methods
," IEEE Trans.