Re: Problems with instabilities in advection

> Hello all,
>
>I am dealing with a simple 1-D advection equation >simulation propagation of solute in a watershed:
>
>d(Acv)/dx+d(Ac)/dt=S
>
>where A is the cross section of the river (then A(x,t)), >v is the velocity of river flow (then v(x,t)) and c is >concentration. S is a source term.
>Obviuosly, A and V are know for each time step and in the >whole space.
>I used a Lax-Wendroff scheme, but since it is a centered >scheme I used a FTBS on the last cell of my network.
>>From what I know I only need a BC @ x=0 and not @ x=end, >however I have instabilities propagating backwards coming >out of watershed outlet, and I assumed it is due to a BC >problem.
>Do I need a particular or different numerical boundary >condition at the outlet?
>
>Thanx!
>
>Alessandro


Hi Alessandro,

you should start with the simplest possible upwind-scheme:

if v > 0, 0 = x_1 < x_2 < ... < x_(N-1) < x_N = xend
is an equidistant discretisation of the spatial domain
with step size dx, and the stepsize in time is dt,
you can calculate

(Ac)_j^(n+1) =
(Ac)_j^n - dt/dx * ((Acv)_j^n- (Acv)_(j-1)^n) + S_j^n * dt

for 2 <= j <= N and n = 1,2,3,...

(perhaps first by setting the source term to zero).

Because (Ac)_j^0 (initial condition) is known
for j=1,...,N , and
(Ac)_1^n is given for all times t_n
(boundary condition at x=0), you can directly
calculate the values for (Ac)_j^(n+1).

Make the time step small enough to ensure that the
CFL-condition is satisfied.

If this scheme works, you can try a second-order-scheme
to reduce numerical diffusion.

Good luck, and let us know about your progress.

Best wishes
Torsten.
.