Analysis of the linearly implicit mid-point rule for differential-algebraic equations

Claus Schneider

Abstract

The error of the linearly implicit mid–point rule after $2m+1$ steps is expanded in powers of $m^2$. We prove that the well-known expansion for ordinary differential equations (an expansion in negative powers of $m^2$) is perturbed by additional terms with non-negative powers of $m^2$ 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.

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Key words

Differential–algebraic equations, linearly implicit mid–point rule, Rosenbrock–type methods, extrapolation.

AMS subject classifications

65L05, 65B05, 58F99.

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