The 1D convection-diffusion equation is given by: \begin{equation}\tag{1} \frac{\partial u}{\partial t} + v \frac{\partial u}{\partial x} - \mu \frac{\partial^2 u}{\partial x^2} = 0, \end{equation} where $u$ is the solution variable, $v$ is the convection velocity, and $\mu$ is the viscosity coefficient.

  1. Spatial Discretization: Define cell averages: $$\tag{2} U_i = \frac{1}{dx} \int_{x_{i-1/2}}^{x_{i+1/2}} u(x, t) \, dx, $$ where $x_{i-1/2}$ and $x_{i+1/2}$ are the left and right boundaries of cell $i$.

  2. Time Integration: Use a first-order upwind scheme for convection and centered differences for diffusion: $$\tag{3} \frac{U_i^{n+1} - U_i^n}{\Delta t} + v \frac{U_i^n - U_{i-1}^n}{dx} - \mu \frac{U_{i+1}^n - 2U_i^n + U_{i-1}^n}{dx^2} = 0. $$

  3. Solve for $U_i^{n+1}$: $$\tag{4} U_i^{n+1} = U_i^n - v \frac{\Delta t}{dx} (U_i^n - U_{i-1}^n) + \mu \frac{\Delta t}{dx^2} (U_{i+1}^n - 2U_i^n + U_{i-1}^n). $$


I want a similar numerical scheme for 2D convection-diffusion equation that is given as follows $$\tag{5}\frac{\partial u}{\partial t} + v \nabla_{\textbf{x}} u = \mu\Delta u, \quad \textbf{x} = (x,y)^T \in \Omega, \ \ t \in \mathbb{R}^{+}.$$ Where the domain is a region between two non-concentric circles (but the smaller one lies inside of another) and the boundary and inital conditions:

\begin{align}\tag{6} &u(\textbf{x}, t) = 0, \quad \textbf{x} \in \partial\Omega, \ \ t \in \mathbb{R}^+ \\ &u(\textbf{x}, 0) = u_0(\textbf{x}), \quad \textbf{x} \in \Omega \tag{7}.\end{align}

I have triangulated the domain (see the figure which I couldn't resize, sorry).

If I want to discretize the equation (cell-centered discretization), I will need to find the weak form of the problem by integrating over each controll volume (triangle) $A_i$ and use $$\frac{1}{|A_i|}\int_{A_i} v\nabla_{\textbf x} u d\textbf x = \frac{1}{|A_i|}\oint_{\partial A_i} (v\cdot n)u d\textbf s,\tag{8}$$ and $$\frac{1}{|A_i|}\int_{A_i} \mu\Delta u d\textbf x = \frac{\mu}{|A_i|}\oint_{\partial A_i} \nabla u\cdot n d\textbf s.\tag{9}$$ For the time dependent term, we can still use the approach from 1D case $$\frac{\partial u_{i}^n}{\partial t} \approx \frac{U_{i}^{n+1} - U_{i}^{n}}{\Delta t}.\tag{10}$$

However, I am not sure how to efficiently discretize the other integrals. At least, it doesn't seem as simple as the 1D case.

enter image description here



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