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Finite difference grid for discretizing PDE (grid point formulation)

The unit square is now discretized into JMAX by JMAX grid points where the boundary conditions and dependent variables will be evaluated. The grid spacing is δ = 1 / ( J M A X - 1 ) . The first and last row and column are boundary values.

The partial derivatives will be approximated by finite differences. For example, the second derivative of vorticity is discretized by a Taylor's series.

w i + 1 = w i + δ w x i + δ 2 2 2 w x 2 i + δ 3 6 3 w x 3 i + δ 4 24 4 w x 4 x ¯ w i - 1 = w i - δ w x i + δ 2 2 2 w x 2 i - δ 3 6 3 w x 3 i + δ 4 24 4 w x 4 ¯ ¯ x 2 w x 2 i = w i - 1 - 2 w i + w i + 1 δ 2 - δ 2 24 4 w x 4 x ¯ + 4 w x 4 ¯ ¯ x = w i - 1 - 2 w i + w i + 1 δ 2 + O δ 2

The finite difference approximation to the PDE at the interior points results in the following set of equations.

ψ i + 1 , j + ψ i - 1 , j + α 2 ψ i , j + 1 + α 2 ψ i , j - 1 - 2 ( 1 + α 2 ) ψ i , j + δ 2 w i , j = 0 w i + 1 , j + w i - 1 , j + α 2 w i , j + 1 + α 2 w i , j - 1 - 2 ( 1 + α 2 ) w i , j = 0 i , j = 2 , 3 , J M A X - 1

The vorticity at the boundary is discretized and expressed in terms of the components of velocity at the boundary, the stream function values on the boundary and a stream function value in the interior grid. (A greater accuracy is possible by using two interior points.) The stream function at the first interior point ( i = 2 ) from the x boundary is written with a Taylor's series as follows.

ψ 2 = ψ 1 ± δ ψ x 1 + δ 2 2 2 ψ x 2 1 + O ( δ 3 ) 2 ψ x 2 1 = 2 ( ψ 2 - ψ 1 ) δ 2 2 δ ψ x 1 + O δ = 2 ( ψ 2 - ψ 1 ) δ 2 ± 2 δ v y B C + O δ w 1 B C v y x - α v x y B C = - 2 ψ x 2 1 - α v x B C y = - 2 ( ψ 2 - ψ 1 B C ) δ 2 2 δ v y B C - α v x B C y + O δ

The choice of sign depends on whether x is increasing or decreasing at the boundary. Similarly, on a y boundary,

w 1 B C = - 2 α 2 ( ψ 2 - ψ 1 B C ) δ 2 ± 2 α v x B C δ + v y B C x + O δ

The boundary condition on the stream function is specified by the normal component of velocity at the boundaries. Since we have assumed zero normal component of velocity, the stream function is a constant on the boundary, which we specify to be zero.

The stream function at the boundary is calculated from the normal component of velocity by numerical integration using the trapezodial rule, e.g.,

ψ j = ψ j - 1 + δ 2 v x j - 1 + v x j / α

Solution of linear equations

The finite difference equations for the PDE and the boundary conditions are a linear system of equations with two dependent variables. The dependent variables at a x i , y j grid point will be represented as a two component vector of dependent variables,

u i , j = ψ i , j w i , j

The pair of equations for each grid point can be represented in the following form

a i , j u i + 1 , j + b i , j u i - 1 , j + c i , j u i , j + 1 + d i , j u i , j - 1 + e i , j u i , j = f i , j i , j = 2 , 3 , J M A X - 1

Each coefficient is a 2x2 matrix. For example,

e i , j u i , j = e i , j , 1 , 1 ψ i , j + e i , j , 1 , 2 w i , j e i , j , 2 , 1 ψ i , j + e i , j , 2 , 2 w i , j

The components of the 2x2 coefficient matrix are the coefficients from the difference equations. The first row is coefficients for the stream function equation and the second row is coefficients for the vorticity equation. The first column is coefficients for the stream function variable and the second column is the coefficients for the vorticity variable. For example, at interior points not affected by the boundary conditions,

a i , j = 1 0 0 1 b i , j = 1 0 0 1 c i , j = α 2 0 0 α 2 d i , j = α 2 0 0 α 2 e i , j = - 2 ( 1 + α 2 ) δ 2 0 - 2 ( 1 + α 2 ) f i , j = 0 0

The coefficients for the interior grid points adjacent to a boundary are modified as a result of substitution the boundary value of stream function or the linear equation for the boundary vorticity into the difference equations. The stream function equation is coupled to the vorticity with the e i j coefficient and the vorticity equation is coupled to the stream function through the boundary conditions. For example, at a x = 0 boundary, the coefficients will be modified as follows.

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
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Sherica
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Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
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a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
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ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
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Asali
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Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Porter
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Cesar
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what is nano technology
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what is system testing?
AMJAD
preparation of nanomaterial
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
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AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
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Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
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Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
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Azam
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
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