Runge-Kutta method projected

Ascher, U. M., and Petzold, L. R., Projected implicit Runge-Kutta methods for differential algebraic equations, SIAM J. Num. Anal. 28(4), 1097 (1991). 

Fig. 3.11 The radial distribution computed using the projected Runge-Kutta method (3.12)-(3.13) with IM steps of stepsize h = 0.01, in comparison to that obtained using the unprojected RK4 method with the same number of steps. This result demonstrates improving energy conservation can have an important effect on the accuracy of computed averages...

The order of implicit Runge-Kutta methods is different for y and A components in index-2 problems. For the two most important collocation methods we give order results in Tab. 5.4. Applying projection to the Gaufi method in order to force that the numerical result is on the constraint manifold increases the order, so that the accuracy of the y variable corresponds to the order of the method in the explicit ODE case. 

---