How to define fluxes for two dimensional convection-diffusion equation?

Question

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.

Answer 1

Edited after some corrections in the question:

One has to decide at which time level you evaluate the fluxes, i. e. to add also the index $n$ or $n+1$ to the values of $T, F, G$. The scheme is then explicit (in time) or implicit (in time). You took the explicit one. After your corrections the scheme looks like standard Finite Volume Method, see e. g. the book of LeVeque on "FVM for hyperbolic problems". By the way for positive lambda your equation is parabolic.

The scheme is stable in general depending on the time discretization, on how large is $\lambda$ or how fine is your mesh. You may use an upwind scheme for the convective part of your fluxes to improve the stability.

The explicit scheme to be stable you have the restriction on the time step $\delta t $ that must be proportional to $dx^2/\lambda$ or $dy^2/\lambda$ (which one is smaller). If you would choose the implicit scheme with the first order accurate upwind you have no stability restriction on $\delta t$, but you have to solve a system of linear equations with tridiagonal matrix.