# Finite Differencing schemes for Convection-Diffusion equation

I'm using the Convection(/advection)-Diffusion(-Reaction) equation to calculate the temperature over time in different hydraulic parts like a pipe or a heat exchanger.
The flow/convection is always 1D, while the diffusion, in this case, heat conduction, can be 1D, 2D or 3D.
Let's consider an example fully in 1D for the sake of easiness:
$$\frac{\delta T}{\delta t} = \frac{\lambda}{c_p \rho}\frac{\delta^2 T}{\delta x^2} + v\frac{\delta T}{\delta x}$$ $$\alpha = \frac{\lambda}{c_p \rho}$$ where $\lambda$ is the heat conductivity, $c_p$ the specific heat capacity, $\rho$ the density and $v$ the velocity.

To discretize this equation, I use the forward-time central-space scheme for the diffusion (second order central) and the forward-time forward-space scheme for the convection (first order upwind), as shown here: $$\frac{T^{n+1} - T^n}{\Delta t} \approx \alpha \frac{T^n_{i-1} - 2T^n_{i} + T^n_{i+1}}{\Delta x^2} + v \frac{T^n_{i\pm1} - T^n_{i}}{\Delta x}$$ where the sign $\pm$ in $T^n_{i\pm1}$ is depending on the sign of $v$.

Now there are two common cases:

• $v \neq 0$ and the resulting Peclet-Number is (in most cases) $|Pe| > 2$
• $v = 0$ and the resulting Peclet-Number is thus $Pe = 0$ (and the PDE becomes fully parabolic)

Having a relatively small timestep is ok in most cases since time continuous systems like PID controllers interact with the simulation results in between each step. These, for example, control the massflows, internal heat gains etc. Thus I've been using an embedded Heun and/or RK method with adaptive stepsize control to get the explicit time results.

## From this multiple questions arise:

• Is using this kind of mixed discretization ok?
• Should I use central differencing scheme for convection for $|Pe| < 2$
• Could I implement a QUICK-scheme for the convective term to reduce numeric errors?
• Is implementing higher accuracy schemes for these derivative orders, as shown in this table, useful?
• When having mixed kinds of "parts" or parts with a big differences in the size of the grid spacing (and thus of the resulting mass/volume of the cells around each node) this results in the parts with the smallest cells never (or really slow) reaching the temperature of the inflowing 1D massflow (neglecting cell heat loss by conduction). Is this called numeric diffusion or dissipation? And can I reduce this by using other differencing schemes like QUICK?
• I'd like to solve a few parts with implicit schemes, Crank-Nicholson is preferred. Is Crank-Nicholson space discretization centered scheme for both convection AND diffusion or can I use centered scheme for diffusion and upwind/QUICK for convection?

edit: Is there any additional information I could provide to make it easier to answer this question?

• Stability analysis would answer some of your questions. You could use Von-Neumann stability analysis to determine when the mixed discretization is stable (1st question). – Jiaqi Li Aug 7 '18 at 16:50
• Thannks for your answer. I am already checking for von Neumann stability for convection and diffusion and I control the timestep, so that it is always in a stable region. From what I read since I posted this question, mixed discretization seems to be ok and also the most common way to go. Any hints for the other questions? – Scotty1- Aug 8 '18 at 12:00