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Journal articles on the topic 'Linear Multistep Methods'

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Published: 4 June 2021
Last updated: 1 February 2022

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1

Hairer, Ernst. "Symmetric linear multistep methods." BIT Numerical Mathematics 46, no. 3 (August 16, 2006): 515–24. http://dx.doi.org/10.1007/s10543-006-0066-z.

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2

Butcher, J. C., and A. T. Hill. "Linear Multistep Methods as Irreducible General Linear Methods." BIT Numerical Mathematics 46, no. 1 (March 2006): 5–19. http://dx.doi.org/10.1007/s10543-006-0046-3.

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3

Hundsdorfer, Willem, Steven J. Ruuth, and Raymond J. Spiteri. "Monotonicity-Preserving Linear Multistep Methods." SIAM Journal on Numerical Analysis 41, no. 2 (January 2003): 605–23. http://dx.doi.org/10.1137/s0036142902406326.

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4

Sahoo, G., and N. Datta. "Auxiliary linear multistep methods: explicit." International Journal of Computer Mathematics 26, no. 2 (January 1989): 101–15. http://dx.doi.org/10.1080/00207168908803688.

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5

Sahoo, G., and N. Datta. "Auxiliary linear multistep methods: implicit." International Journal of Computer Mathematics 31, no. 1-2 (January 1989): 115–23. http://dx.doi.org/10.1080/00207168908803793.

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6

Oliveira, Paula, and Fernanda Patricio. "Instability in linear multistep methods." Applicable Analysis 28, no. 1 (January 1988): 1–14. http://dx.doi.org/10.1080/00036818808839745.

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7

Sand, Jørgen. "Circle contractive linear multistep methods." BIT 26, no. 1 (March 1986): 114–22. http://dx.doi.org/10.1007/bf01939367.

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8

Boutelje, B. R., and A. T. Hill. "Nonautonomous stability of linear multistep methods." IMA Journal of Numerical Analysis 30, no. 2 (February 20, 2009): 525–42. http://dx.doi.org/10.1093/imanum/drn070.

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9

Lenferink, H. W. J. "Contractivity-preserving implicit linear multistep methods." Mathematics of Computation 56, no. 193 (January 1, 1991): 177. http://dx.doi.org/10.1090/s0025-5718-1991-1052098-0.

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10

Jódar, L., J. L. Morera, and E. Navarro. "On convergent linear multistep matrix methods." International Journal of Computer Mathematics 40, no. 3-4 (January 1991): 211–19. http://dx.doi.org/10.1080/00207169108804014.

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11

Karaa, Samir. "Improved Accuracy of Linear Multistep Methods." Applied Mathematics & Information Sciences 7, no. 2 (March 1, 2013): 491–96. http://dx.doi.org/10.12785/amis/070209.

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12

Lenferink, H. W. J. "Contractivity preserving explicit linear multistep methods." Numerische Mathematik 55, no. 2 (March 1989): 213–23. http://dx.doi.org/10.1007/bf01406515.

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13

Odeh, F., O. Nevanlinna, and W. Liniger. "Multiplier and contractivity methods for linear multistep methods." Applied Numerical Mathematics 5, no. 1-2 (February 1989): 89–103. http://dx.doi.org/10.1016/0168-9274(89)90026-3.

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14

Liu, X., M. H. Song, and M. Z. Liu. "Linear Multistep Methods for Impulsive Differential Equations." Discrete Dynamics in Nature and Society 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/652928.

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This paper deals with the convergence and stability of linear multistep methods for impulsive differential equations. Numerical experiments demonstrate that both the mid-point rule and two-step BDF method are of orderp=0when applied to impulsive differential equations. An improved linear multistep method is proposed. Convergence and stability conditions of the improved methods are given in the paper. Numerical experiments are given in the end to illustrate the conclusion.
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15

TIAN, HONGJIONG, and QIAN GUO. "DYNAMICS OF LINEAR MULTISTEP METHODS FOR DELAY DIFFERENTIAL EQUATIONS." International Journal of Bifurcation and Chaos 14, no. 01 (January 2004): 329–36. http://dx.doi.org/10.1142/s0218127404009132.

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In this paper we study the relationship between the asymptotic behavior of a numerical simulation by linear multistep method and that of the true solution itself for fixed step sizes. The numerical method is viewed as a dynamical system in which the step size acts as a parameter. Numerical stability of linear multistep method for nonlinear delay differential equation is investigated and we prove that A-stable linear multistep methods are NP-stable. It is shown that a consistent zero-stable linear multistep method does not admit spurious fixed points. The existence of spurious period-two solutions in the time-step is also studied. Finally we give a simple example to illustrate instability of the spurious period-two solutions.
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16

de Gee, Maarten. "Linear Multistep Methods for Functional Differential Equations." Mathematics of Computation 48, no. 178 (April 1987): 633. http://dx.doi.org/10.2307/2007833.

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17

Banjai, L., and D. Peterseim. "Parallel multistep methods for linear evolution problems." IMA Journal of Numerical Analysis 32, no. 3 (November 15, 2011): 1217–40. http://dx.doi.org/10.1093/imanum/drq040.

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18

de Gee, Maarten. "Linear multistep methods for functional-differential equations." Mathematics of Computation 48, no. 178 (May 1, 1987): 633. http://dx.doi.org/10.1090/s0025-5718-1987-0878696-1.

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19

Hadjimichael, Yiannis, and David I. Ketcheson. "Strong-stability-preserving additive linear multistep methods." Mathematics of Computation 87, no. 313 (February 20, 2018): 2295–320. http://dx.doi.org/10.1090/mcom/3296.

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20

Mincsovics, Miklós E. "Discrete C1 convergence of linear multistep methods." Journal of Computational and Applied Mathematics 363 (January 2020): 234–40. http://dx.doi.org/10.1016/j.cam.2019.06.001.

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21

van Lith, Bart S., Jan H. M. ten Thije Boonkkamp, and Wilbert L. IJzerman. "Full linear multistep methods as root-finders." Applied Mathematics and Computation 320 (March 2018): 190–201. http://dx.doi.org/10.1016/j.amc.2017.09.003.

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22

Keller, Rachael T., and Qiang Du. "Discovery of Dynamics Using Linear Multistep Methods." SIAM Journal on Numerical Analysis 59, no. 1 (January 2021): 429–55. http://dx.doi.org/10.1137/19m130981x.

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23

Evans, N. Wyn, and Scott Tremaine. "Linear Multistep Methods for Integrating Reversible Differential Equations." Astronomical Journal 118, no. 4 (October 1999): 1888–99. http://dx.doi.org/10.1086/301057.

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24

Hulbert, Gregory M. "Limitations on linear multistep methods for structural dynamics." Earthquake Engineering & Structural Dynamics 20, no. 2 (1991): 191–96. http://dx.doi.org/10.1002/eqe.4290200208.

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25

Liu, X., and Y. M. Zeng. "Linear multistep methods for impulsive delay differential equations." Applied Mathematics and Computation 321 (March 2018): 555–63. http://dx.doi.org/10.1016/j.amc.2017.11.014.

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26

Izzo, Giuseppe, and Zdzislaw Jackiewicz. "Generalized linear multistep methods for ordinary differential equations." Applied Numerical Mathematics 114 (April 2017): 165–78. http://dx.doi.org/10.1016/j.apnum.2016.04.009.

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27

Chawla, M. M., and S. McKee. "Convergence of linear multistep methods with multiple roots." Journal of Computational and Applied Mathematics 14, no. 3 (March 1986): 451–54. http://dx.doi.org/10.1016/0377-0427(86)90079-8.

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28

Bakke, V. L., and Z. Jackiewicz. "Stability analysis of linear multistep methods for delay differential equations." International Journal of Mathematics and Mathematical Sciences 9, no. 3 (1986): 447–58. http://dx.doi.org/10.1155/s0161171286000583.

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Stability properties of linear multistep methods for delay differential equations with respect to the test equationy′(t)=ay(λt)+by(t), t≥0,0<λ<1, are investigated. It is known that the solution of this equation is bounded if and only if|a|<−band we examine whether this property is inherited by multistep methods with Lagrange interpolation and by parametrized Adams methods.
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29

Lastman, G. J., and N. K. Sinha. "Identification of linear continuous-time multivariable systems using linear multistep methods." Electronics Letters 27, no. 2 (1991): 127. http://dx.doi.org/10.1049/el:19910084.

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30

Xu, Y., and J. J. Zhao. "Estimation of Longest Stability Interval for a Kind of Explicit Linear Multistep Methods." Discrete Dynamics in Nature and Society 2010 (2010): 1–18. http://dx.doi.org/10.1155/2010/912691.

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The new explicit linear three-order four-step methods with longest interval of absolute stability are proposed. Some numerical experiments are made for comparing different kinds of linear multistep methods. It is shown that the stability intervals of proposed methods can be longer than that of known explicit linear multistep methods.
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31

Messina, Eleonora, Jacques J. B. de Swart, and Wolter A. van der Veen. "Parallel iterative linear solvers for multistep Runge-Kutta methods." Journal of Computational and Applied Mathematics 85, no. 1 (November 1997): 145–67. http://dx.doi.org/10.1016/s0377-0427(97)00135-0.

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32

in 't Hout, K. J. "On the contractivity of implicit–explicit linear multistep methods." Applied Numerical Mathematics 42, no. 1-3 (August 2002): 201–12. http://dx.doi.org/10.1016/s0168-9274(01)00151-9.

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33

Frank, J., W. Hundsdorfer, and J. G. Verwer. "On the stability of implicit-explicit linear multistep methods." Applied Numerical Mathematics 25, no. 2-3 (November 1997): 193–205. http://dx.doi.org/10.1016/s0168-9274(97)00059-7.

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34

Arnold, Andrea, Daniela Calvetti, and Erkki Somersalo. "Linear multistep methods, particle filtering and sequential Monte Carlo." Inverse Problems 29, no. 8 (July 15, 2013): 085007. http://dx.doi.org/10.1088/0266-5611/29/8/085007.

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35

Dimarco, Giacomo, and Lorenzo Pareschi. "Implicit-Explicit Linear Multistep Methods for Stiff Kinetic Equations." SIAM Journal on Numerical Analysis 55, no. 2 (January 2017): 664–90. http://dx.doi.org/10.1137/16m1063824.

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36

Fatunla, S. O., M. N. O. Ikhile, and F. O. Otunta. "A class of p-stable linear multistep numerical methods." International Journal of Computer Mathematics 72, no. 1 (January 1999): 1–13. http://dx.doi.org/10.1080/00207169908804830.

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37

Mazzia, Francesca, Alessandra Sestini, and Donato Trigiante. "B‐Spline Linear Multistep Methods and their Continuous Extensions." SIAM Journal on Numerical Analysis 44, no. 5 (January 2006): 1954–73. http://dx.doi.org/10.1137/040614748.

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38

Hundsdorfer, Willem, and Steven J. Ruuth. "On monotonicity and boundedness properties of linear multistep methods." Mathematics of Computation 75, no. 254 (November 17, 2005): 655–73. http://dx.doi.org/10.1090/s0025-5718-05-01794-1.

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39

Arévalo, Carmen, Gustaf Söderlind, and José Diaz López. "Constant coefficient linear multistep methods with step density control." Journal of Computational and Applied Mathematics 205, no. 2 (August 2007): 891–900. http://dx.doi.org/10.1016/j.cam.2006.02.060.

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40

Borja, Ronaldo I. "One-step and linear multistep methods for nonlinear consolidation." Computer Methods in Applied Mechanics and Engineering 85, no. 3 (February 1991): 239–72. http://dx.doi.org/10.1016/0045-7825(91)90099-r.

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41

Reyes, J. A., F. García-Alonso, J. M. Ferrándiz, and J. Vigo-Aguiar. "Numeric multistep variable methods for perturbed linear system integration." Applied Mathematics and Computation 190, no. 1 (July 2007): 63–79. http://dx.doi.org/10.1016/j.amc.2007.01.017.

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42

Romero, Ignacio. "Stability analysis of linear multistep methods for classical elastodynamics." Computer Methods in Applied Mechanics and Engineering 193, no. 23-26 (June 2004): 2169–89. http://dx.doi.org/10.1016/j.cma.2004.01.012.

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43

Butusov, Denis, Aleksandra Tutueva, Petr Fedoseev, Artem Terentev, and Artur Karimov. "Semi-Implicit Multistep Extrapolation ODE Solvers." Mathematics 8, no. 6 (June 8, 2020): 943. http://dx.doi.org/10.3390/math8060943.

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Multistep methods for the numerical solution of ordinary differential equations are an important class of applied mathematical techniques. This paper is motivated by recently reported advances in semi-implicit numerical integration methods, multistep and extrapolation solvers. Here we propose a novel type of multistep extrapolation method for solving ODEs based on the semi-implicit basic method of order 2. Considering several chaotic systems and van der Pol nonlinear oscillator as examples, we implemented a performance analysis of the proposed technique in comparison with well-known multistep methods: Adams–Bashforth, Adams–Moulton and the backward differentiation formula. We explicitly show that the multistep semi-implicit methods can outperform the classical linear multistep methods, providing more precision in the solutions for nonlinear differential equations. The analysis of stability regions reveals that the proposed methods are more stable than explicit linear multistep methods. The possible applications of the developed ODE solver are the long-term simulations of chaotic systems and processes, solving moderately stiff differential equations and advanced modeling systems.
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44

Li, Yunfei, and Shoufu Li. "Classical Theory of Linear Multistep Methods for Volterra Functional Differential Equations." Discrete Dynamics in Nature and Society 2021 (March 11, 2021): 1–15. http://dx.doi.org/10.1155/2021/6633554.

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Based on the linear multistep methods for ordinary differential equations (ODEs) and the canonical interpolation theory that was presented by Shoufu Li who is exactly the second author of this paper, we propose the linear multistep methods for general Volterra functional differential equations (VFDEs) and build the classical stability, consistency, and convergence theories of the methods. The methods and theories presented in this paper are applicable to nonneutral, nonstiff, and nonlinear initial value problems in ODEs, Volterra delay differential equations (VDDEs), Volterra integro-differential equations (VIDEs), Volterra delay integro-differential equations (VDIDEs), etc. At last, some numerical experiments verify the correctness of our theories.
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45

Butcher, J. C. "General linear methods." Acta Numerica 15 (May 2006): 157–256. http://dx.doi.org/10.1017/s0962492906220014.

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General linear methods, as multistage multivalue methods, are the natural generalizations of linear multistep and Runge-Kutta methods. This survey contains a discussion of the traditional methods and a motivation for the general linear type of generalization. The new methods are introduced in terms of their formulation and the basic properties of consistency, stability and convergence. The order of general linear methods has to be looked at from a new point of view and it is shown how to use an algebraic structure (equivalent to B-series) to express conditions for a given order. Linear and non-linear stability for the new methods brings the theories for the classical methods into a comprehensive formulation and known results are outlined. Recently a number of subfamilies have been introduced and some of these are considered in detail. This applies in particular to methods with the property known as ‘inherent Runge-Kutta stability’. These seem to have prospects of yielding useful and efficient methods, and some progress towards their practical implementation is outlined. Finally, the relationship between stability functions and order of methods is discussed in a setting wide enough to include general linear methods as well as multiderivative methods, such as Obreshkov methods. The classical barriers due to Ehle, Daniel-Moore and Dahlquist (second barrier) all fit into a common pattern and these are explored in a general setting.
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46

Barrio, Manuel, Kevin Burrage, and Pamela Burrage. "Stochastic linear multistep methods for the simulation of chemical kinetics." Journal of Chemical Physics 142, no. 6 (February 14, 2015): 064101. http://dx.doi.org/10.1063/1.4907008.

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47

Voss, David A., and Mark J. Casper. "Efficient Split Linear Multistep Methods for Stiff Ordinary Differential Equations." SIAM Journal on Scientific and Statistical Computing 10, no. 5 (September 1989): 990–99. http://dx.doi.org/10.1137/0910058.

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48

Arnold, M. "The stabilization of linear multistep methods for constrained mechanical systems." Applied Numerical Mathematics 28, no. 2-4 (October 1998): 143–59. http://dx.doi.org/10.1016/s0168-9274(98)00041-5.

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49

Marciniak, Andrzej, and Malgorzata A. Jankowska. "Interval versions for special kinds of explicit linear multistep methods." Results in Applied Mathematics 6 (May 2020): 100104. http://dx.doi.org/10.1016/j.rinam.2020.100104.

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50

van der Houwen, P. J., and H. J. J. te Riele. "Linear multistep methods for Volterra integral and integro-differential equations." Mathematics of Computation 45, no. 172 (1985): 439. http://dx.doi.org/10.1090/s0025-5718-1985-0804934-5.

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