Predictor-corrector methods for the transport equation

Strong form

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

Linearized weak forms

\[\begin{split} \begin{aligned} &\text{Find} \\ &\textbf{a}^{n}\in V_{\textbf{a}}, \\ &\widetilde{u}^{n+1}\in V_{u}, \\ &\widetilde{\textbf{a}}^{n+1}\in V_{\textbf{a}}, \\ &u^{n+1}\in V_{u} \\ &\text{such that} \\ &\mathbb{F}_{\textbf{a}, \widetilde{u},\widetilde{\textbf{a}},u} \begin{cases} F_1(\textbf{a}^{n}, \textbf{v}) = 0 \quad\forall \textbf{v}\in V_{\textbf{a}} \\ F_2(\widetilde{u}^{n+1}, v) = 0 \quad\forall v\in V_u \\ F_3(\widetilde{\textbf{a}}^{n+1}, \textbf{v}) = 0 \quad\forall \textbf{v}\in V_{\textbf{a}} \\ F_4(u^{n+1}, v) = 0 \quad\forall v\in V_u \\ \end{cases} \end{aligned} \end{split}\]