Continuous Galerkin Petrov Time Discretization Scheme for the Solutions of the Chen System

2015 ◽  
Vol 10 (6) ◽  
Author(s):  
S. Hussain ◽  
Z. Salleh
Keyword(s):  
Fourth Order ◽  
Three Dimensional ◽  
Time Discretization ◽  
Dimensional System ◽  
Numerical Comparison ◽  
Chen System ◽  
Time Step ◽  
Quadratic Nonlinearities ◽  
Continuous Galerkin ◽  
Three Dimensional System

In this paper, the continuous Galerkin Petrov time discretization (cGP) scheme is applied to the Chen system, which is a three-dimensional system of ordinary differential equations (ODEs) with quadratic nonlinearities. In particular, we implement and analyze numerically the higher order cGP(2)-method which is found to be of fourth order at the discrete time points. A numerical comparison with classical fourth-order Runge–Kutta (RK4) is given for the presented problem. We look at the accuracy of the cGP(2) as the Chen system changes from a nonchaotic system to a chaotic one. It is shown that the cGP(2) method gains accurate results at larger time step sizes for both cases.

Stability and Unstability of Chaos in Stretch-Twist-Fold Flow (STF)

2013 ◽  
Vol 631-632 ◽  
pp. 1436-1440
Author(s):  
Ud Din Khan Shahab ◽  
Yong Lu Shu ◽  
Ud Din Khan Salah
Keyword(s):  
Decomposition Method ◽  
Chaotic Behavior ◽  
Three Dimensional ◽  
Dimensional System ◽  
Adomian’S Decomposition Method ◽  
Matlab Program ◽  
Quadratic Nonlinearities ◽  
Computer Based ◽  
Three Dimensional System ◽  
Made In

In this study, the Adomian's decomposition method (ADM) is applied to the STF system. This method has been tested on the STF system which is a three-dimensional system of ODE with quadratic nonlinearities. A computer based Matlab program has been developed in order to solve the STF system. Chaotic and non-chaotic behavior of the system has been analyzed graphically and finally a comparison as well as accuracy between two systems has been made in two-step sizes with detail.

Inner-product theory calculations for some rational potentials in a three-dimensional system

1996 ◽  
Vol 74 (1-2) ◽  
pp. 4-9
Author(s):  
M. R. M. Witwit
Keyword(s):  
Energy Levels ◽  
Three Dimensional ◽  
Inner Product ◽  
Dimensional System ◽  
Product Technique ◽  
Special Cases ◽  
Wide Range ◽  
Perturbation Parameters ◽  
Three Dimensional System ◽  
Range Of Values

The energy levels of a three-dimensional system are calculated for the rational potentials,[Formula: see text]using the inner-product technique over a wide range of values of the perturbation parameters (λ, g) and for various eigenstates. The numerical results for some special cases agree with those of previous workers where available.

The influence of mean shear on Alfvén-internal wave propagation

1976 ◽  
Vol 15 (2) ◽  
pp. 197-222
Author(s):  
R. J. Hartman
Keyword(s):  
Three Dimensional ◽  
Dimensional System ◽  
Linear Wave ◽  
Mean Flow ◽  
Wave Processes ◽  
Initial Value ◽  
Initial Wave ◽  
Wave Phenomena ◽  
Three Dimensional System

This paper uses the general solution of the linearized initial-value problem for an unbounded, exponentially-stratified, perfectly-conducting Couette flow in the presence of a uniform magnetic field to study the development of localized wave-type perturbations to the basic flow. The two-dimensional problem is shown to be stable for all hydrodynamic Richardson numbers JH, positive and negative, and wave packets in this flow are shown to approach, asymptotically, a level in the fluid (the ‘isolation level’) which is a smooth, continuous, function of JH that is well defined for JH < 0 as well as JH > 0. This system exhibits a rich complement of wave phenomena and a variety of mechanisms for the transport of mean flow kinetic and potential energy, via linear wave processes, between widely-separated regions of fluid; this in addition to the usual mechanisms for the absorption of the initial wave energy itself. The appropriate three-dimensional system is discussed, and the role of nonlinearities on the development of localized disturbances is considered.

scholarly journals Discrete-to-continuum modelling of weakly interacting incommensurate two-dimensional lattices

2018 ◽  
Vol 474 (2209) ◽  
pp. 20170612 ◽  
Author(s):  
Malena I. Español ◽  
Dmitry Golovaty ◽  
J. Patrick Wilber
Keyword(s):  
Continuum Model ◽  
Domain Walls ◽  
Three Dimensional ◽  
Variational Model ◽  
Dimensional System ◽  
Two Dimensional ◽  
Starting Point ◽  
Continuum Modelling ◽  
Three Dimensional System ◽  
The Continuum

In this paper, we derive a continuum variational model for a two-dimensional deformable lattice of atoms interacting with a two-dimensional rigid lattice. The starting point is a discrete atomistic model for the two lattices which are assumed to have slightly different lattice parameters and, possibly, a small relative rotation. This is a prototypical example of a three-dimensional system consisting of a graphene sheet suspended over a substrate. We use a discrete-to-continuum procedure to obtain the continuum model which recovers both qualitatively and quantitatively the behaviour observed in the corresponding discrete model. The continuum model predicts that the deformable lattice develops a network of domain walls characterized by large shearing, stretching and bending deformation that accommodates the misalignment and/or mismatch between the deformable and rigid lattices. Two integer-valued parameters, which can be identified with the components of a Burgers vector, describe the mismatch between the lattices and determine the geometry and the details of the deformation associated with the domain walls.

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