Symmetric one-step methods and extrapolation technique for index 1 differential-algebraic systems

Gennady Yu. Kulikov

Faculty of Mechanics and Mathematics, Ulyanovsk State University, L. Tolstoy Str. 42, 432700 Ulyanovsk, Russia

KulikovGYu@ulsu.ru

Contributed talk

In the paper we consider index 1 differential-algebraic systems of the form

$$x'(t)=g\bigl(x(t),y(t)\bigr),\quad y(t)=f\bigl(x(t),y(t)\bigr), \eqno(1)$$

where $t \in [0,T]$, $x(t) \in \mbox {\bf R}^m$, $y(t) \in \mbox {\bf R}^n$, $g:D \subset \mbox {\bf R}^{m+n} \to {\bf R}^m$, $f:D \subset \mbox {\bf R}^{m+n} \to {\bf R}^n$, and the initial conditions $x(0)=x^0, y(0)=y^0$ are consistent; i. e., $y^0=f(x^0,y^0)$. To solve problem (1) numerically, we apply an one-step method and obtain the discrete system which is iterated further by Newton-type (or simple) iterations.

Our first goal is to discuss the global error expansion of the discrete approximation. Following the ideas of [1], we come to the concepts of adjointness and symmetry for one-step methods applied to problem (1). We introduce the adjoint and symmetric one-step methods explicitly that gives a very simple test to identify such methods in practice (for details, see [2]). As a result, we describe symmetric Runge-Kutta formulas.

At present extrapolation methods are probably the most efficient technique for solving both ordinary differential equations and differential-algebraic ones numerically. They allow for the automatic determination of both stepsize and order during the computation. Thus, the last aim of the paper is practical implementation of the extrapolation process for one-step methods applied to problems (1). The main difficulty here is how to control the iterative error coming from the iterative method involved in solving the discrete system. Since the iterative error may influence the asymptotic expansion of the global error significantly, just making the extrapolation impossible (some results concerning implicit extrapolation methods for ordinary differential equations are presented in [3]).

One-step methods possessing a quadratic asymptotic expansion of the global error (it is an expansion with only even order terms) are of special interest. In this case we deal with quadratic extrapolation. This term means that after extrapolation each row of the extrapolation table increases the order of the extrapolation method by two. That can reduce computing time dramatically. However, Kulikov shown in [2] and [3] that among one-step methods only symmetric ones possess the quadratic expansion of the global error and there exist no explicit symmetric one-step methods. That is why we have to justified implicit quadratic extrapolation.