Parabolic Equation

Cari u(x,t) yang memenuhi persamaan ParabolikDengan syarat batas u(x,0) = 0 = u(8,t) dan u(x,0) = 4x x2 di x = i : i = 0, 1 , 2 , 3 , 5.

Solution :c2 = 4 , h = 1, k = 1/8

Lab 1 DiscussionIn lab 1 we solved the advection equation:

The first method we tried was the forward Euler method:

Upwind method, CFL=0.9

Whats Going On?AdvectionDiffusionAdd/subtract

Numerical DiffusionThe alebgra shows that the finite difference equation has both an advective term and a diffusive term. It is in fact a better model for:

InstabilityUpwind method, CFL=1.2 (final timstep only)

Lax-Wendroff method, CFL=0.9

Flux LimitersIn the advection equation lets assume v is positive:

Most flux limiters are based on the ratio of the first order fluxes at node i, i.e.:

Heat EquationCari u(x,t) yang memenuhi persamaan parabolicC2 = 4 , h = 1, k = 1/8