Numerical Treatment of Second Order Differential‐Algebraic Systems

Abstract

We consider the numerical treatment of systems of second order differential-algebraic equations (DAEs). The classical approach of transforming a second order system to first order by introducing new variables can lead to difficulties such as an increase in the index or the loss of structure. We show how we can compute an equivalent strangeness-free second order system using the derivative array approach and we present Runge-Kutta methods for the direct numerical solution of second order DAEs.We discuss the numerical solution of the linear second order differential-algebraic initial value problemwhere M, C, K ∈ C(I, C m×n ) and f (t) ∈ C(I, C m ) are sufficiently smooth together with given initial values x 0 ,ẋ 0 ∈ C n . Usually, in the classical theory of ordinary differential equations, high order systems are turned into first order systems by introducing new variables for the derivatives. However, it has been discussed in several publications, see e.g. [1,4], that for differential-algebraic equations (DAEs) this classical approach has to be performed with great care, since it may lead to a number of mathematical difficulties. In [3,5] several examples show that the classical approach of introducing new variables may lead to higher smoothness requirements for the inhomogeneity, corresponding to an increase in the index of the DAE, or may result in a loss of structures in the system. Further, it is shown in [1, 2, 4, 6] that numerical methods can fail. Therefore, the direct discretization and numerical solution of the second order differential-algebraic system can yield better numerical results. The theoretical analysis of linear second order differential-algebraic equations is studied in [3,5], where condensed forms for matrix triple and corresponding invariants under equivalence transformations are derived. One of the main results is the equivalence to a so-called strangeness-free system, that separates the given second order DAE into uncoupled subsystems of second order differential, first order differential and algebraic equations together with some consistency conditions for the right-hand side. The algebraic approach described in [3,5] gives the theoretical analysis of second order DAEs including decisions about existence and uniqueness of solutions and consistency of initial values, but it cannot be used for the development of numerical methods as it is not feasible for numerical computations. Here, we introduce a numerical computable way for the determination of the characteristic invariants of a given DAE, i.e., the number of second order and the number of first order differential equations, the number of algebraic equations and the number of consistency conditions in the strangeness-free system, as well as for the extraction of an equivalent strangeness-free second order differential-algebraic system. The basic idea is to differentiate the DAE (1) a number of times and to put the original DAE together with the derivatives up to a sufficiently high orderμ into an enlarged system. In this way we obtain t...

Index Reduction for Second Order Singular Systems of Difference Equations

“…By using the algebraic approach, we have analyzed the solvability of second order SiDEs/descriptor systems, based on the derived condensed forms constructed under certain constant rank assumptions. In comparison to the previously known procedures [17,22], we have reduced the number of constant rank conditions in every index reduction step from seven to five. This would enlarge the domain of application for SiDEs (and also for DAEs).…”

“…As have been fully discussed in [13,17] for continuous-time systems, these disadvantages include: (1st) increase the index of the singular system, and therefore the complexity of a numerical method to solve it; (2nd) increase the computational effort due to the bigger size of a new system; (3rd) affect the controllability/observability of the corresponding descriptor system since there exist situations where a new system is uncontrollable while the original one is. Therefore, the algebraic approach, which treats the system directly without reformulating it, has been presented in [13,17,22,23] in order to overcome the disadvantages mentioned above. Nevertheless, even for second order SiDEs, this method has not yet been considered.…”

Questions concerning differential-algebraic operators: Toward a reliable direct numerical treatment of differential-algebraic equations

“…We finish by mentioning that in [15,16] the given DAE is transformed via derivative arrays to a so-called strangeness-free mixed-order system which then can be handled by standard methods. Under additional quite special conditions, such a system is provided by evaluating involved matrix polynomials in [4].…”

A derivative array approach for linear second order differential-algebraic systems

“…However, since the standard way to obtain a strangeness-free first order formulation-first introducing new variables for the derivatives to transform the system into a first order system and then applying the usual index reduction procedures to the first order system-can fail due to a possible increase in the index, at first an index reduction of the higher order system should be used, which is followed by an appropriate order reduction to obtain a suitable strangeness-free first order formulation. Recently, it has been shown in [14,17] that also the direct discretization of the second order system may yield better numerical results and is able to prevent certain numerical difficulties as the failure of numerical methods; see also [1,2,16].…”