How to code a 1D ODE in MATLAB

Dear Community

I am trying to code the following 1D ODE using backward difference approximation:

dW/dt =U* (dW/dX) +C

I have entered the equation in MATLAB as below:

W(1) = some constant; Boundary condition at inlet 
for i = 2:n
    DWDt(i) = - U*((W(i)-W(i-1))/(X(i)-X(i-1)))+C ;     
end

My question is: Although in my equation U*(dW/dX) is a positive term, why I need to use negative sign in the for loop? if I make the term U8 (dW/dX) positive in the for loop, the ODE23 command can't solve the equation.

I would highly appreciate if someone can answer.

[JUBAIR]

Here U is the veloctiy term in m/s

Answer 1

[Torsten]

Usually, the equation reads

dW/dt + U*dW/dx = C      (1)

If U in this equation is positive, your flow goes from left to right, if U is negative, your flow goes from right to left.

Depending on the sign of U in (1), you must discretize dW/dX in point X(i) as

(W(i)-W(i-1))/(X(i)-X(i-1)) if U > 0  (information comes from left)

and as

(W(i+1)-W(i))/(X(i+1)-X(i)) if U < 0 (information comes from right)
    

We call it "Upwind Differencing". Using a discretization that doesn't follow the physical direction of informational flow leads to an unstable discretization and thus a failure of the ODE integrator.

[JUBAIR]

Thanks for the explanation. That helps a lot.