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Factored, a-stable, linear multistep methods
ACM SIGNUM Newsletter, 1979Historically, the development and analysis of methods for ordinary differential equations (ODEs) have been more advanced than those for partial differential equations (PDEs). The present state of numerical methods is no exception; therefore, it behooves the numerical analyst to exploit sophisticated ODE methods for the numerical solution of PDEs.
R. F. Warming, Richard M. Beam
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Nonautonomous stability of linear multistep methods
IMA Journal of Numerical Analysis, 2009A linear scalar nonautonomous initial-value problem (IVP) is governed by a scalar lambda(t) with a nonpositive real part. For a wide class of linear multistep methods, including BDF4-6, it is shown that negative real lambda(t) may be chosen to generate instability in the method when applied to the IVP.
B. R. Boutelje, A. T. Hill
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On the convergence of advanced linear multistep methods
BIT, 1979A convergence theorem is given showing that zero-stable advanced linear multistep methods with orderp consistency have orderp convergence.
McKee, S., Pitcher, N.
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Linear Multistep Methods for Volterra Integro-Differential Equations
Journal of the ACM, 1969The Dahlquist stability analysis for ordinary differential equations is extended to the case of Volterra integro-differential equations. Thus the standard multistep methods can be generalized to furnish algorithms for solving integro-differential equations. Special starting procedures are discussed, and some numerical examples are presented.
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Variable Stepsizes in Symmetric Linear Multistep Methods
2001It is well known the great deal of advantages of integrating reversible systems with symmetric methods. The correct qualitative behaviour is imitated, which leads also to quantitative advantageous properties with respect to the errors and their growth with time.
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A criterion forA(α)-stability of linear multistep methods
BIT, 1969Some easy-to-check conditions are given which together are sufficient forA(α)-stability. As an application the greatest α's are determined which giveA(α)-stability for the differentiation formulae withk=3, 4, 5 and 6.
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Discrete \(C^1\) convergence of linear multistep methods
J. Comput. Appl. Math., 2020zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimation
SIAM Journal on Numerical Analysis, 2022Qiang Du, Haizhao Yang
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