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Preprint 02-2015

Index preserving polynomial representation of nonlinear differential-algebraic systems

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Author(s) : Benjamin Unger , Volker Mehrmann

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 02-2015

MSC 2000

34A09 Implicit equations, differential-algebraic equations
65L80 Methods for differential-algebraic equations

Abstract :
Recently in (Gu, 2011) a procedure was presented that allows to reformulate nonlinear ordinary differential equations in a way that all the nonlinearities become polynomial on the cost of increasing the dimension of the system. We generalize this procedure (called `polynomialization') to systems of differential-algebraic equations (DAEs). In particular, we show that if the original nonlinear DAE is regular and strangeness-free (i.\,e., it has differentiation index one) then this property is preserved by the polynomial representation. For systems which are not strangeness-free, i.\,e., where the solution depends on derivatives of the coefficients and inhomogeneities, we also show that the index is preserved for arbitrary strangeness index. However, to avoid ill-conditioning in the representation one should first perform an index reduction on the nonlinear system and then construct the polynomial representations. Although the analytical properties of the polynomial reformulation are very appealing, care has to be given to the numerical integration of the reformulated system due to additional errors. We illustrate our findings with several examples.

Keywords : differential-algebraic equation, strangeness index, differentiation index, polynomial representation of nonlinear differential-algebraic system, polynomialization, index preservation

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