scholarly journals Long-time and unitary properties of semiclassical initial value representations

2004 ◽  
Vol 120 (1) ◽  
pp. 26-30 ◽  
Author(s):  
C. Harabati ◽  
J. M. Rost ◽  
F. Grossmann
Keyword(s):  
Initial Value ◽  
Long Time ◽  
Initial Value Representations

scholarly journals On global dynamics of 2D convective Cahn–Hilliard equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Dynamics ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Initial Value ◽  
Hilliard Equation ◽  
Long Time ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in $H^{4}(\Omega )$ H 4 ( Ω ) when the initial value belongs to $H^{1}(\Omega )$ H 1 ( Ω ) .

A Multiple Interval Chebyshev-Gauss-Lobatto Collocation Method for Ordinary Differential Equations

2016 ◽  
Vol 9 (4) ◽  
pp. 619-639 ◽  
Author(s):  
Zhong-Qing Wang ◽  
Jun Mu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
High Order Accuracy ◽  
Nonlinear Ordinary Differential Equations ◽  
Initial Value ◽  
Order Accuracy ◽  
Long Time

AbstractWe introduce a multiple interval Chebyshev-Gauss-Lobatto spectral collocation method for the initial value problems of the nonlinear ordinary differential equations (ODES). This method is easy to implement and possesses the high order accuracy. In addition, it is very stable and suitable for long time calculations. We also obtain thehp-version bound on the numerical error of the multiple interval collocation method underH1-norm. Numerical experiments confirm the theoretical expectations.

IMPLEMENTING ARBITRARILY HIGH-ORDER SYMPLECTIC METHODS VIA KRYLOV DEFERRED CORRECTION TECHNIQUE

2010 ◽  
Vol 01 (02) ◽  
pp. 277-301 ◽  
Author(s):  
QUANDONG FENG ◽  
JINGFANG HUANG ◽  
NINGMING NIE ◽  
ZAIJIU SHANG ◽  
YIFA TANG
Keyword(s):  
Hamiltonian Systems ◽  
Gaussian Quadrature ◽  
Numerical Procedure ◽  
Original System ◽  
Algebraic Equations ◽  
Initial Value ◽  
Deferred Correction ◽  
Correction Technique ◽  
Long Time ◽  
Spectral Deferred Correction

In this paper, an efficient numerical procedure is presented to implement the Gaussian Runge–Kutta (GRK) methods (also called Gauss methods). The GRK technique first discretizes each marching step of the initial value problem using collocation formulations based on Gaussian quadrature. As is well known, it preserves the geometric structures of Hamiltonian systems. Existing analysis shows that the GRK discretization with s nodes is of order 2s, A-stable, B-stable, symplectic and symmetric, and hence "optimal" for solving initial value problems of general ordinary differential equations (ODEs). However, as the unknowns at different collocation points are coupled in the discretized system, direct solution of the resulting algebraic equations is in general inefficient. Instead, we use the Krylov deferred correction (KDC) method in which the spectral deferred correction (SDC) scheme is applied as a preconditioner to decouple the original system, and the resulting preconditioned nonlinear system is solved efficiently using Newton–Krylov schemes such as Newton–GMRES method. The KDC accelerated GRK methods have been applied to several Hamiltonian systems and preliminary numerical results are presented to show the accuracy, stability, and efficiency features of these methods for different accuracy requirements in short- and long-time simulations.

Initial Value Problems in Viscoelasticity

1988 ◽  
Vol 41 (10) ◽  
pp. 371-378 ◽  
Author(s):  
W. J. Hrusa ◽  
J. A. Nohel ◽  
M. Renardy
Keyword(s):  
Classical Solutions ◽  
Linear Wave ◽  
Time Behavior ◽  
Initial Value ◽  
One Dimensional ◽  
Nonlinear Viscoelastic ◽  
Linear Wave Propagation ◽  
Long Time ◽  
Nonlinear Viscoelastic Materials ◽  
Time Existence

We review some recent mathematical results concerning integrodiff erential equations that model the motion of one-dimensional nonlinear viscoelastic materials. In particular, we discuss global (in time) existence and long-time behavior of classical solutions, as well as the formation of singularities in finite time from smooth initial data. Although the mathematical theory is comparatively incomplete, we make some remarks concerning the existence of weak solutions (i e, solutions with shocks). Some relevant results from linear wave propagation will also be discussed.

Global Well-Posedness for the Defocusing, Cubic, Nonlinear Wave Equation in Three Dimensions for Radial Initial Data in .H s × .H s-1, s> 1/2

2018 ◽  
Vol 2019 (21) ◽  
pp. 6797-6817
Author(s):  
Benjamin Dodson
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Three Dimensions ◽  
Well Posedness ◽  
Critical Space ◽  
Initial Value ◽  
Long Time ◽  
Well Posed

Abstract In this paper we study the defocusing, cubic nonlinear wave equation in three dimensions with radial initial data. The critical space is $\dot{H}^{1/2} \times \dot{H}^{-1/2}$. We show that if the initial data is radial and lies in $\left (\dot{H}^{s} \times \dot{H}^{s - 1}\right ) \cap \left (\dot{H}^{1/2} \times \dot{H}^{-1/2}\right )$ for some $s> \frac{1}{2}$, then the cubic initial value problem is globally well-posed. The proof utilizes the I-method, long time Strichartz estimates, and local energy decay. This method is quite similar to the method used in [11].

PREFACTOR-FREE FORWARD-BACKWARD SEMICLASSICAL DYNAMICS

2003 ◽  
Vol 02 (03) ◽  
pp. 419-437 ◽  
Author(s):  
YUN-AN YAN ◽  
JIUSHU SHAO
Keyword(s):  
Differential Expression ◽  
Quantum Dynamics ◽  
Numerical Calculations ◽  
Initial Value ◽  
Semiclassical Dynamics ◽  
Semiclassical Approximations ◽  
Initial Value Representations

Within a differential expression of the Heisenberg operator, the forward and backward evolution can be joined together along a closed time contour. This manipulation leads to a dramatic cancellation of oscillations due to the two individual propagators in the Heisenberg operator and the resulting forward-backward propagator is more tractable to semiclassical approximations. This article gives a detailed description of the forward-backward semiclassical dynamics (FBSD) formalism. The semiclassical propagators, especially those of the initial value representations (IVRs), are briefly discussed. The derivation of the FBSD based on the Herman–Kluk propagator is reviewed. Different FBSD formulations with other semiclassical IVRs are worked out and numerical calculations show that they are also capable of describing quantum dynamics semiquantitatively and all display accuracy similar to the classical Wigner method.

Inertial Effects in Poroelasticity

1983 ◽  
Vol 50 (2) ◽  
pp. 334-342 ◽  
Author(s):  
R. M. Bowen ◽  
R. R. Lockett
Keyword(s):  
Equations Of Motion ◽  
Compressive Load ◽  
Elastic Solid ◽  
Impervious Surface ◽  
Free Solution ◽  
Inertial Effects ◽  
Initial Value ◽  
Time Approximation ◽  
Long Time ◽  
Bounded Below

The dynamic behavior of a chemically inert, isothermal mixture of an isotropic elastic solid and an elastic fluid is studied. Geometrically, this mixture is assumed to comprise a layer of fixed depth, bounded below by a rigid, impervious surface, and above by a free surface to which loads are applied. The resulting boundary-initial value problem is solved by use of a Green’s function. Two different loading conditions are used to demonstrate the effect of including inertia terms in the equations of motion. In the first example of a constant compressive load, our result is found to agree with the inertia-free solution only for a certain long-time approximation. The second example shows that for a harmonically varying compression, resonance displacements occur at certain loading frequencies, whereas the solution obtained by neglecting inertia does not predict this behavior.

scholarly journals Quantum and semiclassical phase-space dynamics of a wave packet in strong fields using initial-value representations

2014 ◽  
Vol 16 (10) ◽  
pp. 103040 ◽  
Author(s):  
C Zagoya ◽  
J Wu ◽  
M Ronto ◽  
D V Shalashilin ◽  
C Figueira de Morisson Faria
Keyword(s):  
Phase Space ◽  
Wave Packet ◽  
Initial Value ◽  
Strong Fields ◽  
Space Dynamics ◽  
Initial Value Representations
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