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, 1 , 2 , 3 ,… 5.

Solution : c 2 = 4 , h = 1, k = 1/8

Lab 1 Discussion
• In lab 1 we solved the advection equation:

 u t
 v
 u
 x

0
• The first method we tried was the forward
Euler method: u n

1 j
 u n j
 v
 t h
( u n j
 u n j

1
)

Upwind method, CFL=0.9

What’s Going On?
u n j

1  u n j
 t u n j

1  u n j u n j
 t

1  u n j
 t
 v u n j
 u n j

1 h

 v u n j

1
 u n j

1
 u n
2 j h
Add/subtract j n

1

1

2 u n j
 u n j

1

 v u n j

1
 u n j

1
2 h
 vh
2 u n j

1

2 u n j h
2
 u n j

1

0
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
 t
 v
 u
 x

K
 2 u
 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: 
 t u
 v

 u x

0
• Most flux limiters are based on the ratio of the first order fluxes at node i, i.e.: r i

1 / 2
 u u i i


1 u
 i

1 u i