MA/CS471 Lecture 8 Fall 2003 Prof. Tim Warburton [email protected].
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Transcript of MA/CS471 Lecture 8 Fall 2003 Prof. Tim Warburton [email protected].
![Page 2: MA/CS471 Lecture 8 Fall 2003 Prof. Tim Warburton timwar@math.unm.edu.](https://reader036.fdocuments.us/reader036/viewer/2022082817/56649e4f5503460f94b46d6d/html5/thumbnails/2.jpg)
Today
Today we are going to discuss implementation of the simplestpossible partial differential equations:
Example domain discretization (L-shape domain with 3 cells):
0a bt x y
a
b
![Page 3: MA/CS471 Lecture 8 Fall 2003 Prof. Tim Warburton timwar@math.unm.edu.](https://reader036.fdocuments.us/reader036/viewer/2022082817/56649e4f5503460f94b46d6d/html5/thumbnails/3.jpg)
Traveling Wave Solutions in 1D and 2D
This PDE is the two-dimensional analogue to the PDE we saw last time:
0a bt x y
2D
0at x
1D
Solutions:1
f t xa
2 2
ax byf t
a b
t=0:Later t:
2 2
1 a
ba b
![Page 4: MA/CS471 Lecture 8 Fall 2003 Prof. Tim Warburton timwar@math.unm.edu.](https://reader036.fdocuments.us/reader036/viewer/2022082817/56649e4f5503460f94b46d6d/html5/thumbnails/4.jpg)
In Words
• The initial condition is translated with velocity:•
• i.e. the density does not change shape – it simply translates with a constant velocity.
2 2
1 a
ba b
![Page 5: MA/CS471 Lecture 8 Fall 2003 Prof. Tim Warburton timwar@math.unm.edu.](https://reader036.fdocuments.us/reader036/viewer/2022082817/56649e4f5503460f94b46d6d/html5/thumbnails/5.jpg)
Conservation Law
• We first divide the full domain into quadrilateral cells. For each cell e the following conservation law holds:
• i.e. the total density change in a cell e is equal to the flux of density through the boundary of e
• -- or – the rate of change of material in the cell e is equal to the amount “translated” through the boundary.
4
, ,
1
, , , ,f
e f e fx y
fe e
dx y t dxdy an bn x y t ds
dt
![Page 6: MA/CS471 Lecture 8 Fall 2003 Prof. Tim Warburton timwar@math.unm.edu.](https://reader036.fdocuments.us/reader036/viewer/2022082817/56649e4f5503460f94b46d6d/html5/thumbnails/6.jpg)
Boundary Conditions
• Where do we need to apply boundary conditions:
• Hint – which way is the solution translating?
2 2
1 a
ba b
![Page 7: MA/CS471 Lecture 8 Fall 2003 Prof. Tim Warburton timwar@math.unm.edu.](https://reader036.fdocuments.us/reader036/viewer/2022082817/56649e4f5503460f94b46d6d/html5/thumbnails/7.jpg)
Boundary Conditions
• Where do we need to apply external boundary conditions ?:
• i.e. wherever where is the outwards facing normal for the cell e at the face f :
, , 0e f e fx yan bn
,
,
e fx
e fy
n
n
e,1
,1
ex
ey
n
n
,2
,2
ex
ey
n
n
,4
,4
ex
ey
n
n
,3
,3
ex
ey
n
n
2 2
1 a
ba b
![Page 8: MA/CS471 Lecture 8 Fall 2003 Prof. Tim Warburton timwar@math.unm.edu.](https://reader036.fdocuments.us/reader036/viewer/2022082817/56649e4f5503460f94b46d6d/html5/thumbnails/8.jpg)
Definition of Cell Average and Area
4, ,
1
4, ,
1
1Define : , , ,
:
Then , , , ,
, ,
f
f
e
e
e
e
e
e f e fx y
fe e
e f e fx ye
f e
t x y t dxdyA
A dxdy
dx y t dxdy an bn x y t ds
dt
dan bn x y t ds
dA
t
![Page 9: MA/CS471 Lecture 8 Fall 2003 Prof. Tim Warburton timwar@math.unm.edu.](https://reader036.fdocuments.us/reader036/viewer/2022082817/56649e4f5503460f94b46d6d/html5/thumbnails/9.jpg)
Use Upwinding At Boundary of e
e,f e,
4, ,
1
4, ,
1
e,f e,fx y
fx y
,
,
,
,, ,
1 if n n 0:
0 otherwise
Define
The
n
:
n
,
n 1
,f
f
f
e f e fe e x y
f e
e f e fx y e
f e
e
e
e f
e
e f
f
e f e f e f
dA an bn x y t ds
dt
an bn t d
S ds
a b
s
b Sa
4
1f
• The tau variable acts as a switch.• If tau=1 at a face then rho is approximated by the local cell average at the face • If tau=0 at a face then rho is approximated by the neighbor cell average.
![Page 10: MA/CS471 Lecture 8 Fall 2003 Prof. Tim Warburton timwar@math.unm.edu.](https://reader036.fdocuments.us/reader036/viewer/2022082817/56649e4f5503460f94b46d6d/html5/thumbnails/10.jpg)
Upwind Switch
• In this case so
• i.e. for the surface flux term we should use the cell average density from the neighbor cell.
rhoe,1+
rhoea
b
,1 ,1 0e ex yan bn
,1
,1
ex
ey
n
n
e,1 e,1x y
,1
1 if n n 0:
0 otherwise
0
e
a b
![Page 11: MA/CS471 Lecture 8 Fall 2003 Prof. Tim Warburton timwar@math.unm.edu.](https://reader036.fdocuments.us/reader036/viewer/2022082817/56649e4f5503460f94b46d6d/html5/thumbnails/11.jpg)
Euler Forward In Time
1
1 4e,f e
,,x y1
,,f
,
Define
Then
Hence the sc e
1
h me:
n n
ne
e f e f
e
n ne e
e e
n nn ne e
e fe e e ff
ndt
dA
dt dt
A a bdt
S
• For each cell we now have a discrete space in time and space which will compute approximations the cell average density at a given time level.• We do need to specify an initial value at each cell for the cell average density• We also need to specify boundary conditions at “inflow” edges.
![Page 12: MA/CS471 Lecture 8 Fall 2003 Prof. Tim Warburton timwar@math.unm.edu.](https://reader036.fdocuments.us/reader036/viewer/2022082817/56649e4f5503460f94b46d6d/html5/thumbnails/12.jpg)
Summary of Scheme
30
, ,1
41 e,f e,f
x y,
1, ,,
Set up the initial condition
10 , , 0
3
Iterate for neccessary time st
1
eps:
n n
e e e v e vv
n n n ne e e e f
fe f e
ef
e f
t x y t
dt a bA
S
![Page 13: MA/CS471 Lecture 8 Fall 2003 Prof. Tim Warburton timwar@math.unm.edu.](https://reader036.fdocuments.us/reader036/viewer/2022082817/56649e4f5503460f94b46d6d/html5/thumbnails/13.jpg)
4, ,
1
4, ,
1
4, ,
1
4
, , , ,
,, ,, ,
1
1
,
,
1
, , , ,
,
: 1
,
1
f
f
e f e fx y
fe e
e f e fx y
f e
e f e f
n ne
x yf
e f e f e e f e f
e fe f ee f
e fx y e f e
n ne e
e
e ff e
e
e f
dx y t dxdy an bn x y t ds
dt
dan bn x y t ds
dt
an bn
San bn
Adt
S
Sdt
A
,
4
, ,1
1 en n
e f e e ff e
fA
The time rate ofchange of total density in the cell e
The flux through the boundary of the four faces of cell e
![Page 14: MA/CS471 Lecture 8 Fall 2003 Prof. Tim Warburton timwar@math.unm.edu.](https://reader036.fdocuments.us/reader036/viewer/2022082817/56649e4f5503460f94b46d6d/html5/thumbnails/14.jpg)
Implementation
What do we need for the implementation:
(1) A list of vertex locations (x0,y0,x1,y1…)
(2) For each cell e a list of the four vertices
(3) A routine to calculate the area of each cell {Ae}
(4) A routine to calculate the length of each cell edge {Ae,f}
(5) A routine to calculate the outwards facing normal to each cell edge
(6) A routine to calculate the inflow switch tau for each cell edge.
(7) A routine to calculate the initial density profile.
(8) A routine to calculate dt in the following way
(9) A routine to figure out which cells connect to each cell.
30
, ,1
41 e,f e,f
x y ,, ,1
,
10 , , 0
1
3
Loop:
n n
e e e
e f
v e vv
n n n ne e e e fe
e
f e
ff
S
t x y t
dt a bA
, ,,e f e fx yn n
,e f
, , 0x y t
, ,,,
1 1min
4e
e f e fe fx y e f
Adt
an bn S
![Page 15: MA/CS471 Lecture 8 Fall 2003 Prof. Tim Warburton timwar@math.unm.edu.](https://reader036.fdocuments.us/reader036/viewer/2022082817/56649e4f5503460f94b46d6d/html5/thumbnails/15.jpg)
Details
1) Compute cell area by dividing each cell into two triangles, find their areas, and sum up.
2) Do not assume cells are right-angled quadrilaterals (can be deformed).
3) To compute normal to a face:
,1
,1
e
e
x
y
,2
,2
e
e
x
y
,1
,2 ,1
,1 2 2,2 ,1
,2 ,1 ,2 ,1
1e
e ex
ee ey
e e e e
y yn
x xn x x y y
![Page 16: MA/CS471 Lecture 8 Fall 2003 Prof. Tim Warburton timwar@math.unm.edu.](https://reader036.fdocuments.us/reader036/viewer/2022082817/56649e4f5503460f94b46d6d/html5/thumbnails/16.jpg)
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Project 2: continued
• Create a serial version of the 2d finite volume scheme for the above one way wave equation.
• Make it parallel.
![Page 17: MA/CS471 Lecture 8 Fall 2003 Prof. Tim Warburton timwar@math.unm.edu.](https://reader036.fdocuments.us/reader036/viewer/2022082817/56649e4f5503460f94b46d6d/html5/thumbnails/17.jpg)
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