3
The choice of $k$ is restricted also by the discretization of the source term. To see it, rewrite your scheme to
\begin{equation}
u_m^{n+1} = \left(1 - \frac{k(1-x_m)}{h} - k(1-x_m)\right) u_m^n + \frac{k (1-x_m)}{h} u_{m+1}^n \,.
\end{equation}
You need
$$
1 - \frac{k(1-x_m)}{h} - k(1-x_m) \ge 0
$$
for all $x_m$. Taking $x_m=0$ (the worst case scenario) you ...
2
I've identified a few "problems":
Your analytical solution isn't quite correct. The correct analytical solution to the "infinite domain" advection equation is supposed to be $u(t,x) = u_0(x-at)$. Because you have periodic boundaries, you need to properly account for this by making sure $x-at$ is properly wrapped.
I replaced your ...
1
I found a few issues with your implmentation:
you should replace $t$ by $dt$ in your discrete equation (both in your code and in the question), otherwise it makes no sense !
the second part of your equation (with $a$) seems incomplete or wrong, and there's a mix between $F$ and $f$. Moreover, you flux evaluation is wrong: Fplus should read 0.5*Up**2 for ...
1
It turns out that the main problem here was that I thought just naively extending the 1D advection
$$ \rho^{n+1}_{i} = \rho^n_{i} + u\frac{\Delta t}{\Delta x}(\rho^n_{i-1/2} - \rho^n_{i+1/2}) $$
to 2D just by adding the $y$ term like this:
$$ \rho^{n+1}_{i,j} = \rho^n_{i,j} + u\frac{\Delta t}{\Delta x}(\rho^n_{i-1/2,j} - \rho^n_{i+1/2,j}) + v\frac{\Delta t}...
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