I am running a model I created that includes some sediment transport. I am using the advection equation with a source. The equation I am using is: $$ \frac{\partial C}{\partial t} + u\frac{\partial C}{\partial x} = s $$ In this case I thought that to have stability I needed my courant number $u \Delta t/\Delta x$ to be less than 1, but this is not occurring in my case. Can the source term affect this?
I am using an upwind scheme discretization in space.
This paper presents (Theorem 2.1) the CFL condition $$c \frac{\Delta t}{\Delta x} \leq 0.5 \tag{16}$$ assuming that $s = s(u) \in C^\infty$ and that there exist a (large) constant $C$ and a positive constant $c_1$ such that $$ s(u) \leq -c_1 u^{2k - 1} \quad \text{if } u \geq C \quad \forall k \in 1, 2, \dots \tag{15}$$
Based on their given estimate $$\sum_i \Big( u_i^{(n + 1)} \Big)^2 \leq \sum_i \Big( u_i^{(n)} + C\Big)^2 \tag{17}$$ you might have for very large $C$ indeed a "blow-up" of the solution. However, they also state that for the upwind scheme you get bounded variation for $\boldsymbol{u}^{(0)} \in H^1$ $$\sum_i \Big( u_i^{(n + 1)} - u_{i-1}^{(n + 1)} \Big)^2 \leq C \sum_i \Big( u_i^{(0)} - u_{i-1}^{(0)}\Big)^2 \tag{18}$$ which implies strong convergence of the numerical solution: $\boldsymbol{u}_\Delta \rightarrow \boldsymbol{u}$.