Carmen Arévalo

A collocation formulation of multistep methods for variable step-size extensions

Multistep methods are classically constructed by specially designed difference operators on an equidistant time grid. To make them practically useful, they have to be implemented by varying the step-size according to some error-control algorithm. It is well known how to extend Adams and BDF formulas to a variable step-size formulation. In this paper we present a collocation approach to construct variable step-size formulas. We make use of piecewise polynomials to show that every k-step method of order k + 1 has a variable step-size polynomial collocation formulation.

Improving the accuracy of BDF methods for index 3 differential-algebraic equations

Methods for solving index 3 DAEs based on BDFs suffer a loss of accuracy when there is a change of step size or a change of order of the method. A layer of nonuniform convergence is observed in these cases, andO(1) errors may appear in the algebraic variables. From the viewpoint of error control, it is beneficial to allow smooth changes of step size, and since most codes based on BDFs are of variable order, it is also of interest to avoid the inaccuracies caused by a change of order of the method. In the case of BDFs applied to index 3 DAEs in semi-explicit form, we present algorithms that correct toO(h) the inaccurate approximations to the algebraic variables when there are changes of step size in the backward Euler method. These algorithms can be included in an existing code at a very small cost. We have also described how to obtain formulas that correct theO(1) errors in the algebraic variables appearing after a change of order.

Stabilized multistep methods for index 2 Euler-Lagrange DAEs

We consider multistep discretizations, stabilized by β-blocking, for Euler-Lagrange DAEs of index 2. Thus we may use “nonstiff” multistep methods with an appropriate stabilizing difference correction applied to the Lagrangian multiplier term. We show that orderp =k + 1 can be achieved for the differential variables with orderp =k for the Lagrangian multiplier fork-step difference corrected BDF methods as well as for low orderk-step Adams-Moulton methods. This approach is related to the recently proposed “half-explicit” Runge-Kutta methods.

Convergence of multistep discretizations of DAEs

Standard ODE methods such as linear multistep methods encounter difficulties when applied to differential-algebraic equations (DAEs) of index greater than 1. In particular, previous results for index 2 DAEs have practically ruled out the use of all explicit methods and of implicit multistep methods other than backward difference formulas (BDFs) because of stability considerations. In this paper we embed known results for semi-explicit index 1 and 2 DAEs in a more comprehensive theory based on compound multistep and one-leg discretizations. This explains and characterizes the necessary requirements that a method must fulfill in order to be applicable to semi-explicit DAEs. Thus we conclude that the most useful discretizations are those that avoid discretization of the constraint. A freer use of e.g. explicit methods for the non-stiff differential part of the DAE is then possible.

Unitary partitioning in general constraint preserving DAE integrators

A number of numerical algorithms have been developed for various special classes of DAEs. This paper describes a new variable step size, constraint preserving integrator for general nonlinear fully implicit higher index DAEs. Numerical implementation issues are discussed. Numerical examples illustrate the effectiveness of the new method.

Regular and singular b-blocking of difference corrected multistep methods for nonstiff index-2 DAEs

There are several approaches to using nonstiff implicit linear multistep methods for solving certain classes of semi-explicit index 2 DAEs. Using β-blocked discretizations (Arevalo et al., 1996) Adams-Moulton methods up to order 4 and difference corrected BDF (Soderlind, 1989) methods up to order 7 can be stabilized. As no extra matrix computations are required, this approach is an alternative to projection methods.Here we examine some variants of β-blocking. We interpret earlier results as regular β-blocking and then develop singular β-blocking. In this nongeneric case the stabilized formula is explicit, although the discretization of the DAE as a whole is implicit. We investigate which methods can be stabilized in a broad class of implicit methods based on the BDF ρ polynomials. The class contains the BDF, Adams-Moulton and difference corrected BDF methods as well as other high order methods with small error constants. The stabilizing difference operator τ is selected by a minimax criterion for the moduli of the zeros of σ+τ. The class of explicit methods suitable as β-blocked methods is investigated. With singular β-blocking, Adams-Moulton methods up to order 7 can be stabilized with the stabilized method corresponding to the Adams-Bashforth methods. (Less)

Runge-Kutta restarters for multistep methods in presence of frequent discontinuities

Differential equations with discontinuities or differential equations coupled to discrete systems require frequent re-initializations of the numerical solution process. The classical starting process of multistep methods, based on increasing the order in the initialization phase, is computationally expensive when frequent discontinuities occur. Instead we propose to use the stage values or weight vectors of these specially constructed explicit Runge-Kutta methods for starting processes. Two practical examples demonstrate these methods.