Time-step-size-independent conditioning and sensitivity to perturbations in the numerical solution of index three differential algebraic equations

We propose a simple preconditioning for the equations of motion of constrained mechanical systems in index three form. The scaling transformation is applied to the displacement-velocity-multiplier and to the reduced displacement-multiplier forms. The analysis of the transformed system shows that conditioning and sensitivity to perturbations become independent of the time step size, as in the case of well-behaved ordinary differential equations. The new scaling transformation is simple to implement and does not require the rewriting of the system equations as other approaches do. The theoretical analysis is confirmed by numerical examples.