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## Example application Finite Volume Discretization Numerical Methods for PDEs Spring 2007

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### Example application Finite Volume Discretization ... Flow in porous media. Darcy's Law: experimentally derived law relating flow velocity to pressure drop ... – PowerPoint PPT presentation

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Title: Example application Finite Volume Discretization Numerical Methods for PDEs Spring 2007

1
Example application Finite Volume
Discretization Numerical Methods
for PDEs Spring 2007
• Jim E. Jones

2
Flow in porous media
• Darcys Law experimentally derived law relating
flow velocity to pressure drop
• Conservation Law Net flow is balanced by
source/sinks

3
Solving the pressure equation for porous media
flow
• Putting together yields pressure equation
• If K is constant
• Can solve like programming assignment 1
• Use finite difference discretization
• Solve linear system using Gauss-Seidel

4
Finite Volume discretization an alternative to
finite differences
Construct a dual grid by cutting each grid line.
h
5
Finite Volume discretization an alternative to
finite differences
Construct a dual grid by cutting each grid line.
h
6
Finite Volume discretization equations
pi-1,j
pi,j
Each unknown lies at the center of a cell on the
dual grid.
7
Finite Volume discretization equations
Vi,j
pi-1,j
pi,j
Get an equation for i,j by integrating the PDE
over the volume
8
Divergence Theorem
The volume integral of the divergence of a vector
field is equal to the surface integral of its
component in the normal direction
h
V
S
9
Finite Volume discretization equations
Vi,j
pi-1,j
pi,j
Apply divergence theorem.
10
Finite Volume discretization equations
Vi,j
pi-1,j
pi,j
h
11
Finite Volume discretization equations
Vi,j
pi-1,j
pi,j
h
Length of boundary h
12
Finite Volume discretization equations
Vi,j
pi-1,j
pi,j
h

13
Finite Volume discretization equations
Vi,j
pi-1,j
pi,j
14
Finite Volume discretization equations
Vi,j
pi-1,j
pi,j
Get an equation for i,j by integrating the PDE
over the volume
15
Finite Volume discretization equations
Vi,j
pi-1,j
pi,j
Get an equation for i,j by integrating the PDE
over the volume
Integrating source term over the volume
16
Quick Check what if K1?
Same as finite differences for
17
If Ks constant, get matrix from assignment 1,
otherwise
Ki-1
Ki
How do we define KWest at the interface?
18
If Ks constant, get matrix from assignment 1,
otherwise
Ki-1
Ki
How do we define KWest at the interface? Lets
make the normal component of velocity continuous
across the interface.
19
If Ks constant, get matrix from assignment 1,
otherwise
Ki-1
Ki
pi
pWest
pi-1
20
If Ks constant, get matrix from assignment 1,
otherwise
Ki-1
Ki
pi
pWest
pi-1
Equate Vleft and Vright and solve for pWest
21
If Ks constant, get matrix from assignment 1,
otherwise
Ki-1
Ki
pi
pWest
pi-1
22
If Ks constant, get matrix from assignment 1,
otherwise
Ki-1
Ki
pi
pWest
pi-1
The effective diffusion coefficient is the
harmonic average of the 2.
23
Programming Assignment 3
• Will be like assignment 1, except
• The diffusion coefficient will vary and the
discretization will be done by finite volume
method
• Well replace the Gauss-Seidel solver for the
linear system with something more effective
from assignment 1 to solve the PDE

24
Solving the pressure equation for porous media
flow
Keep problem size as input. Im showing a small
problem for illustration.
• Use finite volumes and Gauss-Seidel.

25
Solving the pressure equation for porous media
flow
• Use finite volumes and Gauss-Seidel.

26
Solving the pressure equation for porous media
flow
• Assume that K is defined by a function on x y

27
Solving the pressure equation for porous media
flow
• Use the K value at volume centers in defining the
finite volume equations

28
Upcoming Schedule
March M W 12
14 19 21 24
28
April M W 2
4 9 11 16
18 23 25
• Programming assignment 3 due March 21
• Take home portion of exam handed out March 28
• Take home due and in class exam April 2
• Programming assignment 4 due April 9
• Final Project presentations April 23 25