Methods for Differential Algebraic Equations

There is only one integration algorithm extensively used for DAEs. This is DASSL [2]. It is an extension of Gear’s method and employs the same highly stable backward differentiation formulae. Some implicit algebraic equations can be thought of as the limiting case of explicit differential equations in which an eigenvalue tends to –∞ and the corresponding time constant tends to zero. This implies use of highly stable methods.

The trouble with DAEs is that it is very easy to construct equations which have

• a unique solution,

• a finite number of solutions greater than 1,

• no solution,

• an infinite number of solutions.

For this reason algorithms like DASSL have a very difficult task. Algorithms for ODEs are now highly developed and reliable. Algorithms like DASSL are much newer and cannot be expected to be as reliable.

Experienced modelers instinctively avoid DAEs. It is certainly good practice to minimize implicit equations. However, it is possible to do some clever things with DAEs and their use is being much more common.