I want to solve the following differential equation using control volume approach on a Cartesian mesh: $$\frac{\partial T}{\partial t} + \frac{\partial T}{\partial x} + \frac{\partial T}{\partial y}= \lambda[\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}]$$
Probably, there would be no harm in writing it as ($\lambda$ being contant): $$\frac{\partial T}{\partial t} + \frac{\partial [T - \lambda\frac{\partial T}{\partial x}]}{\partial x} + \frac{\partial [T - \lambda\frac{\partial T}{\partial x}]}{\partial y}= 0$$ For a moment if I define $$F = T - \lambda\frac{\partial T}{\partial x}$$ $$G = T - \lambda\frac{\partial T}{\partial y}$$
Now if I have a control volume around the point (i, j) Can I write the flux F on the right edge i.e. i+$\frac{1}{2}$ as: $$F_{i+\frac{1}{2}, j}^n = \frac{T_{i,j}^n + T_{i+1,j}}{2}^n - \lambda\frac{T_{i+1,j}^n - T_{i,j}^n}{dx}$$
Using this for all other fluxes, I am writing my discretized equation as (In fact, I want a final form of discretized equation to be like this): $$T_{i,j}^{n+1} = T_{i,j}^{n} - \frac{dt}{dx}[{F^n_{i+\frac{1}{2},j} - F^n_{i-\frac{1}{2},j}}] - \frac{dt}{dy}[{G^n_{i,j+\frac{1}{2}} - G^n_{i,j-\frac{1}{2}}}]$$ Is this definition correct or there could be some problems with this? I am not very familiar with the finite volume discretization of such hyperbolic PDEs. Is this method stable and accurate enough? I further want to use mesh refinement schemes.