Partial differential equations (PDEs) of the parabolic type are often solved numerically by the Crank-Nicholson-method. This method can be considered an integration by the trapezoidal rule applied to a system of ordinary differential equations derived from the PDE by spatial discretization. A set of additional algebraic equations is obtained by the spatial discretization of regions to be described by PDEs of the elliptical type as found e.g. in electrical machines. Moreover a supply device like a three-phase ac power-controller for soft-starting and a transmission shafting with torsional elasticity and inertia in a whole electro-mechanical drive have to be modelled by differential and algebraic equations, too. The complete system of equations is therefore of a differential-algebraic type (DAE). Although absolutely stable, the Crank-Nicholson or trapezoidal rule method applied to these DAEs sometimes results in numerical oscillations falsifying the correct solution as shown e.g. in [1]. This "stiffness problem" depends on the eigenvalues of the system in comparison to the time-step. It appears that very small time-constants are often caused by external electric elements, when e.g. a power-electronic valve in an inductive mesh is turned off and becomes highly resistive. The Theta-method for parabolic type PDEs as described in [2] and [3] is a modified trapezoidal rule, where an additional parameter Theta between 0 and 0.5 determines the weighting of the two boundaries of the integrational interval. Theta=0.5 represents the trapezoidal rule, whereas Theta=0 results in the backward Euler-formula. This method can overcome the stiffness problem. However, its uncritical transfer onto arbitrary DAEs can lead to unacceptably errornous solutions concerning the damping of natural oscillations. This is shown by a theoretical investigation and some simulations of an induction-motor drive with a soft-start-supply and a load inertia driven via a torque-measuring-shaft. From the results the recommendation is derived to handle stiffness problems rather by a modelling closer to physics and the actual technical design, i.e. additionally taking into account parasitic or otherwise neglected elements similar to a regularization of DAEs [4] than by choosing a parameter Theta very less than 0.5.

[1] Belmans, R., Hameyer K. and Mertens, R.: Combined Time-Harmonic-Transient Approach to Calculate the Steady-State Behaviour of Induction Machines, IEEE Transactions on Magnetics, 1999.
[2] Krawczyk, A., Tegopoulos, J. A.: Numerical Modelling of Eddy Currents, Oxford:Clarendon Press, 1993.
[3] Zlamal, M.: Finite element method in heat conduction problems. In: The mathematics of finite elements and applications, Academic Press, New York (1976).
[4] Feldmann, U., Günther M.: Some Remarks on Regularization of Circuit Equations, Conference Proceedings of X. ISTET, 1999.