Eulers Method and Adams-Moulton for DAEs

When applying the implicit Euler method to the DAE (5.1.2) the derivatives y tn) are replaced by a backward difference yn — yn-i)lh. This results in a nonlinear system of equations which has to be solved for obtaining yn and An  

We saw that the equations of motion can be formulated as an index-3 system by using constraints on position level. By using constraints on velocity or A level the equations are obtained in index-2 or index-1 form, respectively. The scheme (5.2.1) is formally applicable to each of these formulations though there are significant differences in performance and accuracy. First, we will demonstrate this by integrating the mathematical pendulum formulated in Cartesian coordinates as DAE. 

With the implicit Euler method the results given in Tab. 5.1 were obtained. 

This table also includes results for the problem formulated in its state space form (ssf) 

Comparing after 2 periods, a,tt = 4.0 s, the absolute error in position Cp, in velocity 6v and in the Lagrange multipliers we see that for all step sizes the index-2 formulation and the explicit ODE formulation (ssf) give the best results while the index-3 and index-1 approach may even fail. It can also be seen from this table that the effort for solving the nonlinear system varies enormously with the index of the problem. In this experiment the nonlinear system was solved by Newton s method, which was iterated until the norm of the increment became less than 10 . If this margin was not reached within 10 iterates the Jacobian was updated. The process was regarded as failed, if even with an updated Jacobian this bound was not met. We see, that the index-3 formulation requires the largest amount of re-evaluations of the Jacobian (NJC) and fails for small step sizes. It is a typical property of 

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