scholarly journals Interval Versions of Central-Difference Method for Solving the Poisson Equation in Proper and Directed Interval Arithmetic

2013 ◽  
Vol 38 (3) ◽  
pp. 193-206 ◽  
Author(s):  
Tomasz Hoffmann ◽  
Andrzej Marciniak ◽  
Barbara Szyszka
Keyword(s):  
Poisson Equation ◽  
Interval Arithmetic ◽  
Difference Method ◽  
Floating Point ◽  
Rounding Errors ◽  
Central Difference ◽  
Interval Method ◽  
The Poisson Equation ◽  
Central Difference Method ◽  
The One

Abstract To study the Poisson equation, the central-difference method is often used. This method has the local truncation error of order O(h2 +k2), where h and k are mesh constants. Using this method in conventional floating-point arithmetic, we get solutions including the method, representation and rounding errors. Therefore, we propose interval versions of the central-difference method in proper and directed interval arithmetic. Applying such methods in floating-point interval arithmetic allows one to obtain solutions including all possible numerical errors. We present numerical examples from which it follows that the presented interval method in directed interval arithmetic is a little bit better than the one in proper interval arithmetic, i.e. the intervals of solutions are smaller. It appears that applying both proper and directed interval arithmetic the exact solutions belong to the interval solutions obtained.

scholarly journals Frequency-Dependent FDTD Algorithm Using Newmark’s Method

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Bing Wei ◽  
Le Cao ◽  
Fei Wang ◽  
Qian Yang
Keyword(s):  
Differential Equation ◽  
Electric Field ◽  
Field Intensity ◽  
Time Domain ◽  
Difference Method ◽  
Electric Field Intensity ◽  
Polarization Vector ◽  
Solution Stability ◽  
Central Difference ◽  
Central Difference Method

According to the characteristics of the polarizability in frequency domain of three common models of dispersive media, the relation between the polarization vector and electric field intensity is converted into a time domain differential equation of second order with the polarization vector by using the conversion from frequency to time domain. Newmarkβγdifference method is employed to solve this equation. The electric field intensity to polarizability recursion is derived, and the electric flux to electric field intensity recursion is obtained by constitutive relation. Then FDTD iterative computation in time domain of electric and magnetic field components in dispersive medium is completed. By analyzing the solution stability of the above differential equation using central difference method, it is proved that this method has more advantages in the selection of time step. Theoretical analyses and numerical results demonstrate that this method is a general algorithm and it has advantages of higher accuracy and stability over the algorithms based on central difference method.

Numerical Method for Earthquake-Induced Pounding between Girder and Shear Key. I: Theory

2010 ◽  
Vol 34-35 ◽  
pp. 1402-1405
Author(s):  
Wei He
Keyword(s):  
Seismic Response ◽  
Difference Method ◽  
Equations Of Motion ◽  
Earthquake Ground Motion ◽  
Analysis Model ◽  
Central Difference ◽  
Kelvin Model ◽  
Shear Key ◽  
Analytical Perspective ◽  
Central Difference Method

Earthquake ground motion can induce out-of-phase vibrations between girders and shear keys, which can result in impact or pounding. The paper investigated pounding between girder and shear key from an analytical perspective. By introducing the initial gap in the analysis model, the elastomer stiffness played a role in the transverse vibration as well. A simplified model of bridge transverse seismic response considering girder-shear key pounding was developed. The equations of motion of the bridge response to transverse ground excitation were assembled and solved using the central difference method. Pounding was simulated using a contact force-based model—Kelvin model. Thus, the girder-shear key pounding effects and bridge transverse seismic response can be obtained by using a step-by-step direct integration the central difference method with the appropriate parameters. The proposed method is very useful in the seismic design of bridge.

Computational Methods Used in the Determination of Loading Rate: Experimental and Clinical Implications

1999 ◽  
Vol 15 (4) ◽  
pp. 404-417 ◽  
Author(s):  
C. Mark Woodard ◽  
Margaret K. James ◽  
Stephen P. Messier
Keyword(s):  
Difference Method ◽  
Ground Reaction Force ◽  
Loading Rate ◽  
Reaction Force ◽  
Vertical Force ◽  
Vertical Ground Reaction Force ◽  
Central Difference ◽  
Symptomatic Knee Osteoarthritis ◽  
Central Difference Method ◽  
Vertical Ground

Our purpose was to compare methods of calculating loading rate to the first peak vertical ground reaction force during walking and provide a rationale for the selection of a loading rate algorithm in the analysis of gait in clinical and research environments. Using vertical ground reaction force data collected from 15 older adults with symptomatic knee osteoarthritis and 15 healthy controls, we: (a) calculated loading rate as the first peak vertical force divided by the time from touchdown until the first peak; (b) calculated loading rate as the slope of the least squares regression line using vertical force and time as the dependent and independent variables, respectively; (c) calculated loading rate over discrete intervals using the Central Difference method; and (d) calculated loading rate using vertical force and lime data representing 20% and 90% of the first peak vertical force. The largest loading rate, which may be of greatest clinical importance, occurred when loading rates were calculated using the fewest number of data points. The Central Difference method appeared to maximize our ability to detect differences between healthy and pathologic cohorts. Finally, there was a strong correlation between methods, suggesting that all four methods are acceptable. However, if maximizing the chances of detecting differences between groups is of primary importance, the Central Difference method appears superior.

scholarly journals A nonlinear equation for ionic diffusion in a strong binary electrolyte

2010 ◽  
Vol 466 (2119) ◽  
pp. 2145-2154 ◽  
Author(s):  
Sandip Ghosal ◽  
Zhen Chen
Keyword(s):  
Poisson Equation ◽  
Characteristic Length ◽  
Nonlinear Partial Differential Equation ◽  
Debye Length ◽  
Ambipolar Diffusion ◽  
One Dimensional ◽  
The Poisson Equation ◽  
Local Charge ◽  
Asymptotic Parameter ◽  
The One

The problem of the one-dimensional electro-diffusion of ions in a strong binary electrolyte is considered. The mathematical description, known as the Poisson–Nernst–Planck (PNP) system, consists of a diffusion equation for each species augmented by transport owing to a self-consistent electrostatic field determined by the Poisson equation. This description is also relevant to other important problems in physics, such as electron and hole diffusion across semiconductor junctions and the diffusion of ions in plasmas. If concentrations do not vary appreciably over distances of the order of the Debye length, the Poisson equation can be replaced by the condition of local charge neutrality first introduced by Planck. It can then be shown that both species diffuse at the same rate with a common diffusivity that is intermediate between that of the slow and fast species (ambipolar diffusion). Here, we derive a more general theory by exploiting the ratio of the Debye length to a characteristic length scale as a small asymptotic parameter. It is shown that the concentration of either species may be described by a nonlinear partial differential equation that provides a better approximation than the classical linear equation for ambipolar diffusion, but reduces to it in the appropriate limit.

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