Advection equation

Strong form

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

Time-discretized weak form

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