B12 - varying diffusion coefficients

source: J. Lee, unpublished (Department of Mathematics, MIT, 2017)

considering a diffusion equation:

\frac{\partial u}{\partial t} = \nabla \cdot (k (x) \nabla \cdot u)

where, $k (x)$ is the diffusivity, and is varying

conservative scheme means that the coefficient of $x_{i}^{l}$ in the calculation of $x_{j}^{l + 1}$ is the same as the coefficient of the point $x_{j}^{l}$ in the calculation of $x_{i}^{l + 1}$ , producing a symmetric matrix
considering it in 1D:

\begin{matrix} \frac{\partial u}{\partial t} = \frac{\partial}{\partial x} (k (x) \frac{\partial u}{\partial x}) \\ \frac{u_{i}^{l + 1} + u_{i}^{l}}{Δ t} = \frac{1}{Δ x^{2}} (k_{i + 1 / 2} (u_{i + 1}^{l} - u_{i}^{l}) - k_{i - 1 / 2} (u_{i}^{l} - u_{i - 1}^{l})) \end{matrix}

this is the 1D conservative scheme
analogously, in 2D:

\begin{matrix} \frac{u_{i, j}^{l + 1} - u_{i, j}^{l}}{Δ t} & = \frac{1}{Δ x^{2}} (k_{i + \frac{1}{2}, j} (u_{i + 1, j}^{l} - u_{i, j}^{l}) - k_{i - \frac{1}{2}, j} (u_{i, j}^{l} - u_{i - 1, j}^{l})) \\ + \frac{1}{Δ y^{2}} (k_{i, j + \frac{1}{2}} (u_{i, j + 1}^{l} - u_{i, j}^{l}) - k_{i, j - \frac{1}{2}} (u_{i, j}^{l} - u_{i, j - 1}^{l})) \end{matrix}