# Simple steady-state advection problem: do I need FVM with upwind scheme?

I have a 2D (x,y) scalar advection problem that describes net blowing snow ($q$) transport at a point. This takes the form

$$q = A - F*\nabla\cdot(q {\bf \hat u}),$$

where A ($kg\cdot m^{-2}\cdot s^{-1}$) is a spatially variable scalar source term, $F$ is fetch (m) and $\hat u$ unit normal parallel to to the wind direction (unitless); $\hat u$ is spatially variable. Initial conditions are [ I believe] $q(x,y)=0$ (no blowing snow, develop steady state) and 0 flux boundary condition. This equation is from the literature, but feels a bit wonky so I'm trying to piece it together.

I would like to solve this via an upwind-scheme -- blowing snow transport moves down wind and should only transport snow to cells downwind.

Further, I would like to solve this on a unstructured triangular mesh.

My hang up (and it's likely a dumb one):

do I discretize this via FVM and then, for each edge, apply an unwind scheme?

Ah the moment I'd like to keep it simple (can move to a better upwind method later), and just find the 'upwind' triangle (via dot-product signs at the edge?) and compute the difference with the current cell. Will that work? If not what is the best way to piece this together?

I appreciate any help, I feel like this is a silly question, but I can't quite wrap my head around the order in which to do this.

• I think you probably want it to be $\partial q/\partial t$ on the left hand side, right? Dec 6 '16 at 16:12
• Also, the equation $\nabla q = \frac{\partial q}{\partial x} + \frac{\partial q}{\partial y}$ is definitely not correct. That's not what the gradient is. Dec 6 '16 at 16:12
• I think the last term in the equation should be $$\nabla\cdot(q \hat{u})$$ right? Regarding the first comment by @WolfgangBangerth, I have the same confusion and this might be partially related to exactly what $q$ is. I first thought it was simply the concentration of snow in the air but your phrase "net blowing snow" makes me unsure. Dec 6 '16 at 16:17
• @Chris -- but the equations make no sense just from a dimensional point of view. $\hat u$ has units meters-per-second, the divergence has units one-over-meter, so the rhs has units [q]-by-second, but the lhs has units [q]. That can't be right. Dec 7 '16 at 0:38
• Yes, it's not a simple advection-reaction equation. Yes, you need upwinding. Dec 7 '16 at 20:19