Behaviour in the Large of Numerical Solutions to One-Dimensional Nonlinear Viscoelasticity by Continuous Time Galerkin Methods

1990 ◽  
Author(s):  
Donald A. French ◽  
Soren Jensen
Keyword(s):  
Continuous Time ◽  
Numerical Solutions ◽  
Nonlinear Viscoelasticity ◽  
Galerkin Methods ◽  
One Dimensional

scholarly journals Tornado-type stationary vortex with nonlinear term due to moisture transport

2010 ◽  
Vol 4 (1) ◽  
pp. 77-82 ◽  
Author(s):  
P. B. Rutkevich ◽  
P. P. Rutkevych
Keyword(s):  
Tropical Cyclones ◽  
Shooting Method ◽  
Nonlinear Term ◽  
Numerical Solutions ◽  
Variable Temperature ◽  
Slow Rotation ◽  
Nonlinear Phenomenon ◽  
One Dimensional ◽  
Stationary Vortex ◽  
General Possibility

Abstract. Tornado vortex is believed to be essentially nonlinear phenomenon; and the puzzle to choose the nonlinear term(s) responsible for its formation is still unresolved. In the present work we consider the nonlinear term associated with atmosphere humidity, by introducing variable temperature gradient depending on the vertical velocity of the fluid. Such term is able to yield energy to the system and is very suitable for such a problem. Other nonlinear terms are neglected, assuming slow rotation, or in other words a "weak" tornado approximation. We consider one-dimensional radial boundary problem, and use a modificaiton of shooting method to satisfy boundary conditions at large radii. Obtained numerical solutions of the nonlinear differential equation qualitatively agree with the observed atmosphere vortices (tornados, tropical cyclones). The obtained results show general possibility of existence of unstable motion even in convectively stable atmosphere stratification.

Diffusion-Limited Cell Dehydration: Analytical and Numerical Solutions for a Planar Model

1999 ◽  
Author(s):  
Alexander V. Kasharin ◽  
Jens O. M. Karlsson
Keyword(s):  
Analytical Solution ◽  
Numerical Solutions ◽  
Planar System ◽  
One Dimensional ◽  
Diffusion Limited ◽  
Dimensional Diffusion ◽  
Constant Diffusivity ◽  
A Cell ◽  
Cell Dehydration ◽  
The One

Abstract The process of diffusion-limited cell dehydration is modeled for a planar system by writing the one-dimensional diffusion-equation for a cell with moving, semipermeable boundaries. For the simplifying case of isothermal dehydration with constant diffusivity, an approximate analytical solution is obtained by linearizing the governing partial differential equations. The general problem must be solved numerically. The Forward Time Center Space (FTCS) and Crank-Nicholson differencing schemes are implemented, and evaluated by comparison with the analytical solution. Putative stability criteria for the two algorithms are proposed based on numerical experiments, and the Crank-Nicholson method is shown to be accurate for a mesh with as few as six nodes.

scholarly journals Dissociation and recombination in the electrolyte flow model

2021 ◽  
Vol 2090 (1) ◽  
pp. 012076
Author(s):  
A Shobukhov ◽  
H Koibuchi
Keyword(s):  
Flow Model ◽  
Numerical Solutions ◽  
Stable Equilibrium ◽  
Steady State Solution ◽  
System Of Equations ◽  
Constant Flow ◽  
Dimensional Model ◽  
Electrolyte Flow ◽  
One Dimensional ◽  
Dilute Aqueous Solution

Abstract We propose a one-dimensional model for the dilute aqueous solution of NaCl which is treated as an incompressible fluid placed in the external electric field. This model is based on the Poisson-Nernst-Planck system of equations, which also contains the constant flow velocity as a parameter and considers the dissociation and the recombination of ions. We study the steady-state solution analytically and prove that it is a stable equilibrium. Analyzing the numerical solutions, we demonstrate the importance of dissociation and recombination for the physical meaningfulness of the model.

scholarly journals Homogenization for deterministic maps and multiplicative noise

2013 ◽  
Vol 469 (2156) ◽  
pp. 20130201 ◽  
Author(s):  
Georg A. Gottwald ◽  
Ian Melbourne
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Multiplicative Noise ◽  
Stochastic Integrals ◽  
Deterministic System ◽  
One Dimensional ◽  
Stable Lévy Process ◽  
Fast Flow

A recent paper of Melbourne & Stuart (2011 A note on diffusion limits of chaotic skew product flows. Nonlinearity 24 , 1361–1367 (doi:10.1088/0951-7715/24/4/018)) gives a rigorous proof of convergence of a fast–slow deterministic system to a stochastic differential equation with additive noise. In contrast to other approaches, the assumptions on the fast flow are very mild. In this paper, we extend this result from continuous time to discrete time. Moreover, we show how to deal with one-dimensional multiplicative noise. This raises the issue of how to interpret certain stochastic integrals; it is proved that the integrals are of Stratonovich type for continuous time and neither Stratonovich nor Itô for discrete time. We also provide a rigorous derivation of super-diffusive limits where the stochastic differential equation is driven by a stable Lévy process. In the case of one-dimensional multiplicative noise, the stochastic integrals are of Marcus type both in the discrete and continuous time contexts.

Numerical Solutions of Transients in Pneumatic Networks—Transmission-Line Calculations

1968 ◽  
Vol 35 (3) ◽  
pp. 588-595 ◽  
Author(s):  
S. Tsao
Keyword(s):  
Wave Propagation ◽  
Transmission Lines ◽  
Numerical Solutions ◽  
Navier Stokes ◽  
Damping Factor ◽  
Temperature Profiles ◽  
Hypothetical Case ◽  
One Dimensional ◽  
Simplifying Assumptions ◽  
Energy Equations

Equations governing the damped wave propagation along transmission lines are obtained from the Navier-Stokes and energy equations by making certain simplifying assumptions. The flow considered is essentially one-dimensional. However, radial variations of the velocity and temperature profiles must be considered, because the damping factor is directly dependent on them. The equations are integrated by numerical methods. A hypothetical case is computed as an example.

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