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:
u
u
v
0
t
x
• The first method we tried was the forward
Euler method:
u
n 1
j
vt n
n
u 
(u j  u j 1 )
h
n
j
Upwind method, CFL=0.9
What’s Going On?
un 1  un
un  un
Add/subtract u nj1
j
j
j
j 1
v

t
h
n  u n   u n  2u n  u n 
n

1
n
u
u
u


j

1
j

1
j

1
j
j

1
j
j


v
t
2h
un 1  un
un  un
u n  2u n  u n
j
j
j 1
j  1 vh j  1
j
j 1
v

0
2
t
2h
2
h
Advection
Diffusion
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:
u
u
u
v
K 2
t
x
x
2
Instability
Upwind method, CFL=1.2 (final timstep only)
Lax-Wendroff method, CFL=0.9
Flux Limiters
• In the advection equation let’s assume v is
positive:
u
u
v
0
t
x
• Most flux limiters are based on the ratio of
the first order fluxes at node i, i.e.:
ui  ui 1
ri 1/ 2 
ui 1  ui