Implementation of Linear Multistep Methods for Solving Constrained Equations of Motion

1993 ◽  
Vol 30 (3) ◽  
pp. 774-789 ◽  
Author(s):  
Florian A. Potra
Keyword(s):  
Multistep Methods ◽  
Equations Of Motion ◽  
Linear Multistep Methods ◽  
Constrained Equations

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.

Investigation of Parametric Resonance Instability in a Flexible Rod Slider Crank Mechanism

1991 ◽  
Author(s):  
David G. Beale ◽  
Shyr-Wen Lee
Keyword(s):  
Potential Energy ◽  
Parametric Resonance ◽  
Variational Approach ◽  
Equations Of Motion ◽  
Monodromy Matrix ◽  
Bending Energy ◽  
Constrained Equations ◽  
Beam Bending ◽  
Floating Frame ◽  
Crank Mechanism

Abstract A direct variational approach with a floating frame is presented to derive the ordinary differential equations of motion of a flexible rod, constant crank speed slider crank mechanism. Potential energy terms contained in the derivation include beam bending energy and energy in foreshortening of the rod tip (which were selected because of the importance of these terms in a pinned-pinned rod parametric resonance). A symbolic manipulator code is used to reduce the constrained equations of motion to unconstrained nonlinear equations. A linearized version of these equations is used to explore parametric resonance stability-instability zones at low crank speeds and small deflections by a monodromy matrix technique.

scholarly journals Trigonometrically-Fitted Methods: A Review

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1197
Author(s):  
Changbum Chun ◽  
Beny Neta
Keyword(s):  
Initial Value Problems ◽  
Polynomial Interpolation ◽  
Multistep Methods ◽  
Error Reduction ◽  
Trigonometric Polynomials ◽  
Linear Multistep Methods ◽  
Initial Value ◽  
First Order ◽  
The Subject ◽  
Trigonometrically Fitted Methods

Numerical methods for the solution of ordinary differential equations are based on polynomial interpolation. In 1952, Brock and Murray have suggested exponentials for the case that the solution is known to be of exponential type. In 1961, Gautschi came up with the idea of using information on the frequency of a solution to modify linear multistep methods by allowing the coefficients to depend on the frequency. Thus the methods integrate exactly appropriate trigonometric polynomials. This was done for both first order systems and second order initial value problems. Gautschi concluded that “the error reduction is not very substantial unless” the frequency estimate is close enough. As a result, no other work was done in this direction until 1984 when Neta and Ford showed that “Nyström’s and Milne-Simpson’s type methods for systems of first order initial value problems are not sensitive to changes in frequency”. This opened the flood gates and since then there have been many papers on the subject.

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