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PPT – Example application Finite Volume Discretization Numerical Methods for PDEs Spring 2007 PowerPoint presentation | free to download - id: fdcfd-YWVhN

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Example application Finite Volume

Discretization Numerical Methods

for PDEs Spring 2007

- Jim E. Jones

Flow in porous media

- Darcys Law experimentally derived law relating

flow velocity to pressure drop - Conservation Law Net flow is balanced by

source/sinks

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

Finite Volume discretization an alternative to

finite differences

Construct a dual grid by cutting each grid line.

h

Finite Volume discretization an alternative to

finite differences

Construct a dual grid by cutting each grid line.

h

Finite Volume discretization equations

pi-1,j

pi,j

Each unknown lies at the center of a cell on the

dual grid.

Finite Volume discretization equations

Vi,j

pi-1,j

pi,j

Get an equation for i,j by integrating the PDE

over the volume

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

Finite Volume discretization equations

Vi,j

pi-1,j

pi,j

Apply divergence theorem.

Finite Volume discretization equations

Vi,j

pi-1,j

pi,j

h

Finite Volume discretization equations

Vi,j

pi-1,j

pi,j

h

Length of boundary h

Finite Volume discretization equations

Vi,j

pi-1,j

pi,j

h

Finite Volume discretization equations

Vi,j

pi-1,j

pi,j

Finite Volume discretization equations

Vi,j

pi-1,j

pi,j

Get an equation for i,j by integrating the PDE

over the volume

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

Quick Check what if K1?

Same as finite differences for

If Ks constant, get matrix from assignment 1,

otherwise

Ki-1

Ki

How do we define KWest at the interface?

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.

If Ks constant, get matrix from assignment 1,

otherwise

Ki-1

Ki

pi

pWest

pi-1

If Ks constant, get matrix from assignment 1,

otherwise

Ki-1

Ki

pi

pWest

pi-1

Equate Vleft and Vright and solve for pWest

If Ks constant, get matrix from assignment 1,

otherwise

Ki-1

Ki

pi

pWest

pi-1

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.

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 - To get a head start, modify your code (or mine)

from assignment 1 to solve the PDE

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.

Solving the pressure equation for porous media

flow

- Use finite volumes and Gauss-Seidel.

Solving the pressure equation for porous media

flow

- Assume that K is defined by a function on x y

Solving the pressure equation for porous media

flow

- Use the K value at volume centers in defining the

finite volume equations

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