Parabolic Equation. Cari u(x,t) yang memenuhi persamaan Parabolik Dengan syarat batas u(x,0) = 0 =...
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Transcript of Parabolic Equation. Cari u(x,t) yang memenuhi persamaan Parabolik Dengan syarat batas u(x,0) = 0 =...
![Page 1: Parabolic Equation. Cari u(x,t) yang memenuhi persamaan Parabolik Dengan syarat batas u(x,0) = 0 = u(8,t) dan u(x,0) = 4x – ½ x 2 di x = i : i = 0,](https://reader035.fdocuments.net/reader035/viewer/2022062320/56649d625503460f94a43ffd/html5/thumbnails/1.jpg)
Parabolic Equation
![Page 2: Parabolic Equation. Cari u(x,t) yang memenuhi persamaan Parabolik Dengan syarat batas u(x,0) = 0 = u(8,t) dan u(x,0) = 4x – ½ x 2 di x = i : i = 0,](https://reader035.fdocuments.net/reader035/viewer/2022062320/56649d625503460f94a43ffd/html5/thumbnails/2.jpg)
![Page 3: Parabolic Equation. Cari u(x,t) yang memenuhi persamaan Parabolik Dengan syarat batas u(x,0) = 0 = u(8,t) dan u(x,0) = 4x – ½ x 2 di x = i : i = 0,](https://reader035.fdocuments.net/reader035/viewer/2022062320/56649d625503460f94a43ffd/html5/thumbnails/3.jpg)
Cari u(x,t) yang memenuhi persamaan Parabolik
Dengan syarat batas u(x,0) = 0 = u(8,t) dan u(x,0) = 4x – ½ x2
di x = i : i = 0, 1 , 2 , 3 ,… 5.
![Page 4: Parabolic Equation. Cari u(x,t) yang memenuhi persamaan Parabolik Dengan syarat batas u(x,0) = 0 = u(8,t) dan u(x,0) = 4x – ½ x 2 di x = i : i = 0,](https://reader035.fdocuments.net/reader035/viewer/2022062320/56649d625503460f94a43ffd/html5/thumbnails/4.jpg)
Solution :
c2 = 4 , h = 1, k = 1/8
![Page 5: Parabolic Equation. Cari u(x,t) yang memenuhi persamaan Parabolik Dengan syarat batas u(x,0) = 0 = u(8,t) dan u(x,0) = 4x – ½ x 2 di x = i : i = 0,](https://reader035.fdocuments.net/reader035/viewer/2022062320/56649d625503460f94a43ffd/html5/thumbnails/5.jpg)
![Page 6: Parabolic Equation. Cari u(x,t) yang memenuhi persamaan Parabolik Dengan syarat batas u(x,0) = 0 = u(8,t) dan u(x,0) = 4x – ½ x 2 di x = i : i = 0,](https://reader035.fdocuments.net/reader035/viewer/2022062320/56649d625503460f94a43ffd/html5/thumbnails/6.jpg)
![Page 7: Parabolic Equation. Cari u(x,t) yang memenuhi persamaan Parabolik Dengan syarat batas u(x,0) = 0 = u(8,t) dan u(x,0) = 4x – ½ x 2 di x = i : i = 0,](https://reader035.fdocuments.net/reader035/viewer/2022062320/56649d625503460f94a43ffd/html5/thumbnails/7.jpg)
![Page 8: Parabolic Equation. Cari u(x,t) yang memenuhi persamaan Parabolik Dengan syarat batas u(x,0) = 0 = u(8,t) dan u(x,0) = 4x – ½ x 2 di x = i : i = 0,](https://reader035.fdocuments.net/reader035/viewer/2022062320/56649d625503460f94a43ffd/html5/thumbnails/8.jpg)
![Page 9: Parabolic Equation. Cari u(x,t) yang memenuhi persamaan Parabolik Dengan syarat batas u(x,0) = 0 = u(8,t) dan u(x,0) = 4x – ½ x 2 di x = i : i = 0,](https://reader035.fdocuments.net/reader035/viewer/2022062320/56649d625503460f94a43ffd/html5/thumbnails/9.jpg)
![Page 10: Parabolic Equation. Cari u(x,t) yang memenuhi persamaan Parabolik Dengan syarat batas u(x,0) = 0 = u(8,t) dan u(x,0) = 4x – ½ x 2 di x = i : i = 0,](https://reader035.fdocuments.net/reader035/viewer/2022062320/56649d625503460f94a43ffd/html5/thumbnails/10.jpg)
![Page 11: Parabolic Equation. Cari u(x,t) yang memenuhi persamaan Parabolik Dengan syarat batas u(x,0) = 0 = u(8,t) dan u(x,0) = 4x – ½ x 2 di x = i : i = 0,](https://reader035.fdocuments.net/reader035/viewer/2022062320/56649d625503460f94a43ffd/html5/thumbnails/11.jpg)
Lab 1 Discussion
• In lab 1 we solved the advection equation:
• The first method we tried was the forward Euler method:
0
x
uv
t
u
)( 11 n
jnj
nj
nj uu
h
tvuu
![Page 12: Parabolic Equation. Cari u(x,t) yang memenuhi persamaan Parabolik Dengan syarat batas u(x,0) = 0 = u(8,t) dan u(x,0) = 4x – ½ x 2 di x = i : i = 0,](https://reader035.fdocuments.net/reader035/viewer/2022062320/56649d625503460f94a43ffd/html5/thumbnails/12.jpg)
Upwind method, CFL=0.9
![Page 13: Parabolic Equation. Cari u(x,t) yang memenuhi persamaan Parabolik Dengan syarat batas u(x,0) = 0 = u(8,t) dan u(x,0) = 4x – ½ x 2 di x = i : i = 0,](https://reader035.fdocuments.net/reader035/viewer/2022062320/56649d625503460f94a43ffd/html5/thumbnails/13.jpg)
What’s Going On?
02
12
122
111
2
12
1111
11
h
njun
jun
juvh
h
njun
juv
t
njun
ju
h
njun
jun
jun
jun
juv
t
njun
ju
h
njun
juv
t
njun
ju
Advection Diffusion
Add/subtract nju 1
![Page 14: Parabolic Equation. Cari u(x,t) yang memenuhi persamaan Parabolik Dengan syarat batas u(x,0) = 0 = u(8,t) dan u(x,0) = 4x – ½ x 2 di x = i : i = 0,](https://reader035.fdocuments.net/reader035/viewer/2022062320/56649d625503460f94a43ffd/html5/thumbnails/14.jpg)
Numerical Diffusion
• The alebgra shows that the finite difference equation has both an advective term and a diffusive term. It is in fact a better model for:
2
2
x
uK
x
uv
t
u
![Page 15: Parabolic Equation. Cari u(x,t) yang memenuhi persamaan Parabolik Dengan syarat batas u(x,0) = 0 = u(8,t) dan u(x,0) = 4x – ½ x 2 di x = i : i = 0,](https://reader035.fdocuments.net/reader035/viewer/2022062320/56649d625503460f94a43ffd/html5/thumbnails/15.jpg)
Upwind method, CFL=1.2 (final timstep only)
Instability
![Page 16: Parabolic Equation. Cari u(x,t) yang memenuhi persamaan Parabolik Dengan syarat batas u(x,0) = 0 = u(8,t) dan u(x,0) = 4x – ½ x 2 di x = i : i = 0,](https://reader035.fdocuments.net/reader035/viewer/2022062320/56649d625503460f94a43ffd/html5/thumbnails/16.jpg)
Lax-Wendroff method, CFL=0.9
![Page 17: Parabolic Equation. Cari u(x,t) yang memenuhi persamaan Parabolik Dengan syarat batas u(x,0) = 0 = u(8,t) dan u(x,0) = 4x – ½ x 2 di x = i : i = 0,](https://reader035.fdocuments.net/reader035/viewer/2022062320/56649d625503460f94a43ffd/html5/thumbnails/17.jpg)
Flux Limiters
• In the advection equation let’s assume v is positive:
• Most flux limiters are based on the ratio of the first order fluxes at node i, i.e.:
0
x
uv
t
u
ii
iii uu
uur
1
12/1