Advection-diffusion equation

Strong form

\[\begin{split} \begin{align*} &\text{Find}~u(\textbf{x}, t): \Omega\times[0,\infty) \to \mathbb{R}~\text{such that } \\ &\mathbb{IBVP}_u\begin{cases} \frac{\partial u}{\partial t} + \textbf{a}\cdot\nabla u= \nabla\cdot(\mathsf{D}\cdot\nabla u) & \forall(\textbf{x}, t)\in\Omega\times[0,\infty) \\ u=u_0 & \forall(\textbf{x},t)\in\Omega\times\{0\}\\ u=u_{\text{D}} & \forall(\textbf{x},t)\in\partial\Omega_{\text{D}}\times[0,\infty) \\ \textbf{n}\cdot(\mathsf{D}\cdot\nabla{u}) = u_{\text{N}} & \forall(\textbf{x},t)\in\partial\Omega_{\text{N}}\times[0,\infty)~,~\partial\Omega_{\text{N}}=\partial\Omega/\partial\Omega_{\text{D}} \end{cases} \\ &\text{given} \\ &\mathbb{S}_u\begin{cases} \Omega\subset\mathbb{R}^d & \text{domain} \\ u_0(\textbf{x}) & \text{initial condition} \\ u_{\text{D}}(\textbf{x}, t)~,~\partial\Omega_{\text{D}} & \text{Dirichlet boundary condition}\\ u_{\text{N}}(\textbf{x}, t)~,~\partial\Omega_{\text{N}} & \text{Neumann boundary condition}\\ \textbf{a}(\textbf{x}, t) & \text{velocity} \\ \mathsf{D}(\textbf{x}, t) & \text{dispersion} \\ \end{cases} \end{align*} \end{split}\]

Time-discretized weak form

\[\begin{split} \begin{aligned} &\text{Find}~u^{n+1}\in V~\text{such that } \\ &\begin{align*} F(u^{n+1},v)&=\int_\Omega\text{d}\Omega~v\frac{u^{n+1} - u^n}{\Delta t^n} + v\mathcal{D}_{{\textbf{a}},u}(\textbf{a}\cdot\nabla u) \\ &\qquad\quad + \nabla v\cdot\mathcal{D}_{{\mathsf{D}},u}(\mathsf{D}\cdot\nabla u) \\ &\quad - \int_{\partial\Omega_{\text{N}}}\text{d}\Gamma~vu_{\text{N}}=0 \quad\forall v\in V \\ \end{align*} \\ &\text{with finite difference operators}~\mathcal{D}_{\mathbf{a},u}, \mathcal{D}_{\mathsf{D},u}. \end{aligned} \end{split}\]