| 1. |
- Arévalo, Carmen, et al.
(författare)
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Regular and singular β-blocking of difference corrected multistep methods for nonstiff index-2 DAEs
- 2000
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Ingår i: Applied Numerical Mathematics. - 0168-9274. ; 35:4, s. 293-305
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Tidskriftsartikel (refereegranskat)abstract
- 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 operatorτ 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.
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| 2. |
- Arévalo, Carmen, et al.
(författare)
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Stabilized multistep methods for index 2 Euler-Lagrange DAEs
- 1996
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Ingår i: BIT. - 0006-3835. ; 36:1, s. 1-13
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Tidskriftsartikel (refereegranskat)abstract
- 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 order p =k + 1 can be achieved for the differential variables with order p =k for the Lagrangian multiplier fork-step difference corrected BDF methods as well as for low order k-step Adams-Moulton methods. This approach is related to the recently proposed “half-explicit” Runge-Kutta methods.
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| 3. |
- Arévalo, Carmen, et al.
(författare)
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β-blocked multistep methods for Euler-Lagrange DAEs: Linear analysis
- 1997
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Ingår i: ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik. - : Wiley. - 0044-2267. ; 77:8, s. 609-617
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Tidskriftsartikel (refereegranskat)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.
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| 4. |
- Modin, Klas, et al.
(författare)
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Geometric Integration of Weakly Dissipative Systems
- 2009
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Ingår i: Numerical Analysis and Applied Mathematics, Vols 1 and 2. - 1551-7616 .- 0094-243X. ; 1168, s. 877-877
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Konferensbidrag (refereegranskat)abstract
- Some problems in mechanics, e.g. in bearing simulation, contain subsystems that are conservative as well as weakly dissipative subsystems. Our experience is that geometric integration methods are often superior for such systems, as long as the dissipation is weak. Here we develop adaptive methods for dissipative perturbations of Hamiltonian systems. The methods are "geometric" in the sense that the form of the dissipative perturbation is preserved. The methods are linearly explicit, i.e., they require the solution of a linear subsystem. We sketch an analysis in terms of backward error analysis and numerical comparisons with a conventional RK method of the same order is given.
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