| 1. |
- Arévalo, Carmen, et al.
(författare)
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A software platform for adaptive high order multistep methods
- 2020
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Ingår i: ACM Transactions on Mathematical Software. - : Association for Computing Machinery (ACM). - 0098-3500 .- 1557-7295. ; 46:1
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Tidskriftsartikel (refereegranskat)abstract
- We present a software package, Modes, offering h-adaptive and p-adaptive linear multistep methods for first order initial value problems in ordinary differential equations. The implementation is based on a new parametric, grid-independent representation of multistep methods [Arévalo and Söderlind 2017]. Parameters are supplied for over 60 methods. For nonstiff problems, all maximal order methods (p=k for explicit and p=k+1 for implicit methods) are supported. For stiff computation, implicit methods of order p=k are included. A collection of step-size controllers based on digital filters is provided, generating smooth step-size sequences offering improved computational stability. Controllers may be selected to match method and problem classes. A new system for automatic order control is also provided for designated families of multistep methods, offering simultaneous h- and p-adaptivity. Implemented as a Matlab toolbox, the software covers high order computations with linear multistep methods within a unified, generic framework. Computational experiments show that the new software is competitive and offers qualitative improvements. Modes is available for downloading and is primarily intended as a platform for developing a new generation of state-of-the-art multistep solvers, as well as for true ceteris paribus evaluation of algorithmic components. This also enables method comparisons within a single implementation environment.
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| 2. |
- Arévalo, Carmen, et al.
(författare)
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Constant coefficient linear multistep methods with step density control
- 2007
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier BV. - 0377-0427. ; 205:2, s. 891-900
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Tidskriftsartikel (refereegranskat)abstract
- In linear multistep methods with variable step size, the method's coefficients are functions of the step size ratios. The coefficients therefore need to be recomputed on every step to retain the method's proper order of convergence. An alternative approach is to use step density control to make the method adaptive. If the step size sequence is smooth, the method can use constant coefficients without losing its order of convergence. The paper introduces this new adaptive technique and demonstrates its feasibility with a few test problems. The technique works in perfect agreement with theory for a given step density function. For practical use, however, the density must be generated with data computed from the numerical solution. We introduce a local error tracking controller, which automatically adapts the density to computed data, and demonstrate in computational experiments that the technique works well at least up to fourth-order methods. (c) 2006 Elsevier B.V. All rights reserved.
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| 3. |
- Arévalo, Carmen, et al.
(författare)
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Convergence of multistep discretizations of DAEs
- 1995
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Ingår i: BIT. - 0006-3835. ; 35:2, s. 143-168
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 4. |
- Arévalo, Carmen, et al.
(författare)
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GRID-INDEPENDENT CONSTRUCTION OF MULTISTEP METHODS
- 2017
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Ingår i: Journal of Computational Mathematics. - : Global Science Press. - 0254-9409 .- 1991-7139. ; 35, s. 672-692
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Tidskriftsartikel (refereegranskat)abstract
- A new polynomial formulation of variable step size linear multistep methods is presented, where each k-step method is characterized by a fixed set of k-1 or k parameters. This construction includes all methods of maximal order (p=k for stiff, and p=k+1 for nonstiff problems). Supporting time step adaptivity by construction, the new formulation is not based on extending classical fixed step size methods; instead classical methods are obtained as fixed step size restrictions within a unified framework. The methods are implemented in Matlab, with local error estimation and a wide range of step size controllers. This provides a platform for investigating and comparing different multistep method in realistic operational conditions. Computational experiments show that the new multistep method construction and implementation compares favorably to existing software, although variable order has not yet been included.
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| 5. |
- Arévalo, Carmen, et al.
(författare)
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Local error estimation and step size control in adaptive linear multistep methods
- 2021
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Ingår i: Numerical Algorithms. - : Springer Science and Business Media LLC. - 1017-1398 .- 1572-9265. ; 86:2, s. 537-563
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Tidskriftsartikel (refereegranskat)abstract
- In a k-step adaptive linear multistep methods the coefficients depend on the k − 1 most recent step size ratios. In a similar way, both the actual and the estimated local error will depend on these step ratios. The classical error model has been the asymptotic model, chp+ 1y(p+ 1)(t), based on the constant step size analysis, where all past step sizes simultaneously go to zero. This does not reflect actual computations with multistep methods, where the step size control selects the next step, based on error information from previously accepted steps and the recent step size history. In variable step size implementations the error model must therefore be dynamic and include past step ratios, even in the asymptotic regime. In this paper we derive dynamic asymptotic models of the local error and its estimator, and show how to use dynamically compensated step size controllers that keep the asymptotic local error near a prescribed tolerance tol. The new error models enable the use of controllers with enhanced stability, producing more regular step size sequences. Numerical examples illustrate the impact of dynamically compensated control, and that the proper choice of error estimator affects efficiency.
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| 6. |
- 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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| 7. |
- 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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| 8. |
- 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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| 9. |
- Babuška, Ivo, et al.
(författare)
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On roundoff error growth in elliptic problems
- 2018
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Ingår i: ACM Transactions on Mathematical Software. - : Association for Computing Machinery (ACM). - 0098-3500 .- 1557-7295. ; 44:3
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Tidskriftsartikel (refereegranskat)abstract
- Large-scale linear systems arise in finite-difference and finite-element discretizations of elliptic problems. With increasing computer performance, ever larger systems are solved using direct methods. How large can such systems be without roundoff compromising accuracy? Here we model roundoff dynamics in standard LU and LDLT decompositions with respect to problem size N. For the one-dimensional (1D) Poisson equation with Dirichlet boundary conditions on an equidistant grid, we show that the relative error in the factorized matrix grows like O(ϵN) if roundoffs are modeled as independent, expectation zero random variables. With bias, the growth rate changes to O(ϵN). Subsequent back substitution results in typical error growths of O(ϵNN) and O(ϵN2), respectively. Error growth is governed by the dynamics of the computational process and by the structure of the boundary conditions rather than by the condition number. Computational results are demonstrated in several examples, including a few fourth-order 1D problems and second-order 2D problems, showing that error accumulation depends strongly on the solution method. Thus, the same LU solver may exhibit different growth rates for the same 2D Poisson problem, depending on whether the five-point or nine-point FDM operator is used.
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| 10. |
- Hairer, E, et al.
(författare)
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Explicit, time reversible, adaptive step size control
- 2005
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 26:6, s. 1838-1851
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Tidskriftsartikel (refereegranskat)abstract
- Adaptive step size control is difficult to combine with geometric numerical integration. As classical step size control is based on "past" information only, time symmetry is destroyed and with it the qualitative properties of the method. In this paper we develop completely explicit, reversible, symmetry-preserving, adaptive step size selection algorithms for geometric numerical integrators such as the Stormer-Verlet method. A new step density controller is proposed and analyzed using backward error analysis and reversible perturbation theory. For integrable reversible systems we show that the resulting adaptive method nearly preserves all action variables and, in particular, the total energy for Hamiltonian systems. It has the same excellent long-term behavior as that obtained when constant steps are used. With variable steps, however, both accuracy and efficiency are greatly improved.
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