Analysis of the Linearly Implicit Mid–point Rule for Differential–algebraic Equations

The error of the linearly implicit mid–point rule after 2m + 1 steps is expanded in powers of m2. We prove that the well-known expansion for ordinary differential equations (an expansion in negative powers of m2) is perturbed by additional terms with non-negative powers of m2 for semi–explicit differential–algebraic equations of index one. Hence, extrapolation in m−2 will be of limited value only. The complete expansion shows these limits and, furthermore, can be used to derive an order 8 method of Rosenbrock type.