How to handle inflow and outflow boundaries for a non-linear convection-diffusion equation (DGFEM)

Question

Following "A conservative DGM for Convection-Diffusion and Navier-Stokes Problems" (Oden and Baumann), if we have a linear convection-diffusion equation of the following form: $$ \nabla\cdot(\mathbf{b}u) - \nabla\cdot(a\nabla u) = f $$

with BC's $$ u = u_{D} \;\;\text{on} \;{\Gamma_{D}} \;\;\;\text{and}\;\;\;a\nabla u\cdot\mathbf{n}=u_{N} \;\;\text{on} \;\Gamma_{N} $$

Then we have the variational formulation of the convection term as follows: $$ A_{\mathbf{b}}(u,v) = -\sum_{K\in\mathcal{M}_{h}}\int_{K}\nabla v\cdot(\mathbf{b}u) + \sum_{e\in\Gamma_{h}^{i}}\int_{e}\left\{\mathbf{b}\cdot\mathbf{n}\right\}u^{\uparrow}[v] + \sum_{e\in\Gamma_{\text{out}}}\int_{e}(\mathbf{b}\cdot\mathbf{n})uv $$

and $$ L_{\mathbf{b}}(v) = -\sum_{e\in\Gamma_{\text{in}}}\int_{e}(\mathbf{b}\cdot\mathbf{n})u_{D}v $$

where $\Gamma_{h}^{i}$ is the set of interior edges, $\Gamma_{\text{in}} = \left\{\mathbf{x}\in\partial\Omega:\;\mathbf{b}\cdot\mathbf{n} < 0\right\}$ and $\Gamma_{\text{out}} = \partial\Omega\backslash\Gamma_{\text{in}}$. Also, we have the following upwind term: $$ u^{\uparrow} = \begin{cases} u|_{K_{-}}, &\left\{\mathbf{b}\cdot\mathbf{n}\right\} \geqslant0\\ u|_{K_{+}}, &\left\{\mathbf{b}\cdot\mathbf{n}\right\} < 0\end{cases} $$

where $K_{-}$ is the element to the "left" of an interior edge $e$ and $K_{+}$ is the element to the "right", using a fixed convention for defining the neighbors of an edge.

So I can implement this without much trouble. However, I'm trying to solve a non-linear equation in the following form: $$ \nabla\cdot(\gamma(u)\mathbf{b}(u)) - \nabla\cdot(a(u)\nabla u) = f(x,y) $$

I'm having some trouble figuring out the convective term here. I have: $$ A_{\mathbf{b}}(u,v) = -\sum_{K\in\mathcal{M}_{h}}\int_{K}\nabla v\cdot(\gamma(u)\mathbf{b}(u)) + \sum_{e\in\Gamma_{h}^{i}}\int_{e}\left\{\mathbf{b}(u)\cdot\mathbf{n}\right\}(\gamma(u))^{\uparrow}[v] + \sum_{e\in\Gamma_{\text{out}}}\int_{e}(\mathbf{b}(u)\cdot\mathbf{n})\gamma(u)v $$

and $$ L_{\mathbf{b}}(u) = -\sum_{e\in\Gamma_{\text{in}}}\int_{e}(\mathbf{b}(u)\cdot\mathbf{n})\gamma(u_{D})v $$

1) I'm trying to compute a Jacobian from these terms and I think there may be something wrong with my integration by parts. Is my averaging and upwinding correct? Namely, I'm a bit unsure of where the $\uparrow$ should go.

2) On the inflow integral, do I evaluate $\mathbf{b}(u_{D})$ AND $\gamma(u_{D})$? Or just $\gamma(u_{D})$? In the first case, this inflow term wouldn't contribute to the Jacobian. In the latter case, it would. Does it matter which way I do it?

Answer 1

I do not quite see why you separate $\gamma(u)$ and $\mathbf b(u)$.

Do the integration by parts, that should yield on interior faces the consistency term \begin{equation} \{\gamma(u)\mathbf b(u)\cdot\mathbf n\}[v] \end{equation} To this, you can add a consistent stabilization term like multiples of $[\gamma(u)\mathbf b(u)\cdot\mathbf n][v]$, possibly of $[\gamma(u)\{\mathbf b(u)\cdot\mathbf n\}[v]$. But only if your interface term is the sum of the consistency term above and a stabilization term which vanishes if applied to continuous functions, will you discretization be conservative.

If this holds, you might still be in trouble with stability. Have you tried local Lax-Friedrichs fluxes?

By the way, you know about the stability issues with the Baumann-Oden method?