scholarly journals The stability problem for linear multistep methods: Old and new results

2007 ◽  
Vol 210 (1-2) ◽  
pp. 2-12 ◽  
Author(s):  
L. Aceto ◽  
D. Trigiante
Keyword(s):  
Stability Problem ◽  
Multistep Methods ◽  
Linear Multistep Methods ◽  
The Stability

scholarly journals On the stability of implicit-explicit linear multistep methods

1997 ◽  
Vol 25 (2-3) ◽  
pp. 193-205 ◽  
Author(s):  
J. Frank ◽  
W. Hundsdorfer ◽  
J.G. Verwer
Keyword(s):  
Multistep Methods ◽  
Linear Multistep Methods ◽  
The Stability

scholarly journals Estimation of Longest Stability Interval for a Kind of Explicit Linear Multistep Methods

2010 ◽  
Vol 2010 ◽  
pp. 1-18 ◽  
Author(s):  
Y. Xu ◽  
J. J. Zhao
Keyword(s):  
Absolute Stability ◽  
Numerical Experiments ◽  
Multistep Methods ◽  
Linear Multistep Methods ◽  
Stability Intervals ◽  
The Stability ◽  
Order Four

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.

An Improved Stiffly Stable Method for Direct Integration of Nonlinear Structural Dynamic Equations

1975 ◽  
Vol 42 (2) ◽  
pp. 464-470 ◽  
Author(s):  
K. C. Park
Keyword(s):  
Multistep Methods ◽  
Nonlinear Problems ◽  
New Method ◽  
Linear Multistep Methods ◽  
Stable Method ◽  
Structural Dynamic ◽  
Linear Problems ◽  
The Stability ◽  
Dynamics Problems ◽  
Derivative Information

The behavior of linear multistep methods has been evaluated for application to structural dynamics problems. By examining the local stability of the currently popular methods as applied to nonlinear problems, it is shown that the presence of historical derivatives can cause numerical instability in the nonlinear dynamics even for methods that are unconditionally stable for linear problems. Through an understanding of the stability characteristics of Gear’s two-step and three-step methods, a new method requiring no historical derivative information has been developed. Superiority of the new method for nonlinear problems is indicated by means of comparisons with currently popular methods.

Symmetric Multistep Methods Revisited: II. Numerical Experiments

1999 ◽  
Vol 173 ◽  
pp. 309-314 ◽  
Author(s):  
T. Fukushima
Keyword(s):  
Multistep Methods ◽  
Computational Time ◽  
Integration Time ◽  
Implicit Methods ◽  
Symmetric Methods ◽  
Celestial Bodies ◽  
The Stability ◽  
Predictor Corrector ◽  
Symmetric Multistep Methods ◽  
Integration Errors

AbstractBy using the stability condition and general formulas developed by Fukushima (1998 = Paper I) we discovered that, just as in the case of the explicit symmetric multistep methods (Quinlan and Tremaine, 1990), when integrating orbital motions of celestial bodies, the implicit symmetric multistep methods used in the predictor-corrector manner lead to integration errors in position which grow linearly with the integration time if the stepsizes adopted are sufficiently small and if the number of corrections is sufficiently large, say two or three. We confirmed also that the symmetric methods (explicit or implicit) would produce the stepsize-dependent instabilities/resonances, which was discovered by A. Toomre in 1991 and confirmed by G.D. Quinlan for some high order explicit methods. Although the implicit methods require twice or more computational time for the same stepsize than the explicit symmetric ones do, they seem to be preferable since they reduce these undesirable features significantly.

scholarly journals The two-dimensional hydrodynamical theory of moving aerofoils. IV

1947 ◽  
Vol 188 (1015) ◽  
pp. 439-463
Keyword(s):  
Stability Problem ◽  
Two Dimensional ◽  
Centre Of Gravity ◽  
First Approximation ◽  
Von Karman ◽  
Moving Cylinder ◽  
Period Equation ◽  
The Stability ◽  
Von Kármán ◽  
Hydrodynamical Theory

In the first part of this paper opportunity has been taken to make some adjustments in certain general formulae of previous papers, the necessity for which appeared in discussions with other workers on this subject. The general results thus amended are then applied to a general discussion of the stability problem including the effect of the trailing wake which was deliberately excluded in the previous paper. The general conclusion is that to a first approximation the wake, as usually assumed, has little or no effect on the reality of the roots of the period equation, but that it may introduce instability of the oscillations, if the centre of gravity of the element is not sufficiently far forward. During the discussion contact is made with certain partial results recently obtained by von Karman and Sears, which are shown to be particular cases of the general formulae. An Appendix is also added containing certain results on the motion of a vortex behind a moving cylinder, which were obtained to justify certain of the assumptions underlying the trail theory.

scholarly journals High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

2021 ◽  
Author(s):  
Giacomo Albi ◽  
Lorenzo Pareschi
Keyword(s):  
Multistep Methods ◽  
General Setting ◽  
Reaction Diffusion ◽  
Strong Stability ◽  
High Order ◽  
Iterative Solvers ◽  
Time Dependent ◽  
Linear Multistep Methods ◽  
Advection Diffusion Equation ◽  
Separation Of Scales

AbstractWe consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta methods, these semi-implicit techniques give a great flexibility, and allow, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work, we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-diffusion equation and in the setting of strong stability preserving (SSP) methods. Our findings are demonstrated on several examples, including nonlinear reaction-diffusion and convection-diffusion problems.

Sign in / Sign up

Export Citation Format

Share Document