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