β‐blocked Multistep Methods for Euler‐Lagrange DAEs: Linear Analysis

Abstract

Many different methods have been suggested for the numerical solution of index 2 and 3 Euler‐Lagrange equations. We focus on 0‐stability of multistep methods (φ, σ) and investigate the relations between some well‐known computational techniques. By various modifications, referred to as β‐blocking of the σ polynomial, some basic shortcomings of multistep methods may be overcome. This approach is related to projection techniques and has a clear and well‐known analogy in control theory. In particular, it is not necessary to use BDF methods for the solution of high index problems; indeed, “nonstiff” methods may be used for part of the system provided that the state‐space form is nonstiff. We illustrate the techniques and demonstrate the results with a simplified multibody model of a truck.

Grid-Independent Construction of Multistep Methods

“…In [3,4] we developed the β-blocking approach to solving differential-algebraic equations (DAEs) using various combinations of multistep methods. New, dedicated families of non-stiff implicit and explicit methods were derived in this context, in particular the implicit and explicit difference correction methods (IDC and EDC, respectively), [4].…”

Stabilized multistep methods for index 2 Euler-Lagrange DAEs

“…The application of methods suitable for nonstiff problems has been studied recently, see e .g . [3,4] . In the case of multistep methods, this entails combining different multistep formulas for different parts A December 1994.…”

“…In [3] we studied how to apply these compound methods to the problem with a linear algebraic constraint . This paper continues the study of convergence for such methods .…”

“…This effectively limits the useful methods to a rather narrow class, with the BDF methods as the most common choice . However, compound discretizations offer a wider range of feasible methods, [1,2,3] . We shall consider the approach introduced in [3] .…”

“…However, compound discretizations offer a wider range of feasible methods, [1,2,3] . We shall consider the approach introduced in [3] . The strong method requirements referred to above are due to the way in which the multiplier appears in (1 .1) .…”