parabolic equation. cari u(x,t) yang memenuhi persamaan parabolik dengan syarat batas u(x,0) = 0 =...
Post on 21-Dec-2015
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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 x2
di x = i : i = 0, 1 , 2 , 3 ,… 5.
Solution :
c2 = 4 , h = 1, k = 1/8
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
Upwind method, CFL=0.9
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
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
Upwind method, CFL=1.2 (final timstep only)
Instability
Lax-Wendroff method, CFL=0.9
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