Spatial discretization of the governing equations

We discretize both the Stokes and heat diffusion equations using a Finite Differences approach on a staggered grid (ref Taras book here).

Heat diffusion

where dotted lines represent the velocity ghost nodes.

We discretize both the Stokes and heat diffusion equations using a Finite Differences approach on a staggered grid (ref Taras book here).

\begin{aligned} \tilde{θ} \frac{q_{x}^{n + 1} - q_{x}^{n}}{Δ ψ} + q_{x}^{n + 1} & = - κ \frac{T^{n} + T^{n}}{Δ x} \\ \tilde{θ} \frac{q_{y}^{n + 1} - q_{y}^{n}}{Δ ψ} + q_{y}^{n + 1} & = - κ \frac{T^{n} + T^{n}}{Δ x} \\ \tilde{ρ} \frac{T^{n + 1} + T^{n}}{Δ ψ} + ρ C_{p} \frac{T^{n + 1} + T^{t}}{Δ t} & = - (\frac{\partial q_{x}}{\partial x} + \frac{\partial q_{y}}{\partial y}) \end{aligned}

Upon convergence we recover

\begin{array}{r} T^{t + Δ t} = T^{n + 1} \\ q_{x}^{t + Δ t} = q_{x}^{n + 1} \\ q_{y}^{t + Δ t} = q_{y}^{n + 1} \end{array}

\begin{array}{r} \tilde{ρ} \frac{u_{x}^{n + 1} - u_{x}^{n}}{Δ ψ} = \nabla \cdot τ - \frac{p^{n + 1} - p^{n}}{Δ x} - ρ g_{x} \end{array}

\begin{array}{r} \tilde{ρ} \frac{u_{y}^{n + 1} - u_{y}^{n}}{Δ ψ} = \nabla \cdot τ - \frac{p^{n + 1} - p^{n}}{Δ y} - ρ g_{y} \end{array}

\begin{array}{r} \frac{1}{\tilde{K}} \frac{p^{n + 1} - p^{n}}{Δ ψ} + (\frac{u_{x}^{n + 1} - u_{x}^{n}}{Δ x} + \frac{u_{y}^{n + 1} - u_{y}^{n}}{Δ y}) = \frac{1}{Δ t} (β (p^{t + Δ t} - p^{t}) + α (T^{t + Δ t} - T^{t})) \end{array}

\begin{array}{r} \frac{1}{2 \tilde{G}} \frac{τ^{n + 1} - τ^{n}}{Δ ψ} + \frac{1}{2 G} \frac{τ^{n + 1} - τ^{t}}{Δ t} + \frac{τ^{n + 1}}{2 η} = \dot{ε} \end{array}