Advection equation

Strong form

\[\begin{split} \begin{align*} &\text{Find}~u(\textbf{x}, t): \Omega\times[0,\infty) \to \mathbb{R}~\text{such that} \\ &\mathbb{IBVP}\begin{cases} \frac{\partial u}{\partial t}+\textbf{a}\cdot\nabla u = 0 & \forall(\textbf{x}, t)\in\Omega\times[0,\infty) \\ u=u_0 & \forall(\textbf{x},t)\in\Omega\times\{0\}\\ u=u_{\text{I}} & \forall(\textbf{x}, t)\in\Omega_{\text{I}}\times[0,\infty)~,~\partial\Omega_{I} = \{\textbf{x}\in\partial\Omega~:~\textbf{n}\cdot\textbf{a}<0\} \end{cases}~. \end{align*} \end{split}\]

Weak form

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

Specification

\[\begin{split}\mathbb{S}\begin{cases} \Omega \\ u_0(\textbf{x}) \\ u_{\text{I}}(\textbf{x}, t) \\ \textbf{a}(\textbf{x}, t) \\ \end{cases}\end{split}\]