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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?

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  • $\begingroup$ 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). $\endgroup$ – Jiaqi Li Aug 7 '18 at 16:50
  • $\begingroup$ 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? $\endgroup$ – Scotty1- Aug 8 '18 at 12:00
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  1. it is ok for the discretization.
  2. CD scheme has some stability problem when Pe>2, but we can decrease the mesh spacing to obtain a low mesh Pe number.
  3. QUICK-scheme is more stable and accurate than CD, so it is ok to implement it.
  4. you should increase the order of both convection term and diffusion term to obtain a high order solution.
  5. the mesh spacings should be changed gradually. I think you met some problems about the stability.
  6. yes, both convection and diffusion terms can be treated implicitly.
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  • $\begingroup$ Hi ztdep, thanks alot for your help. Considering 2.: Decreasing mesh-spacing is unluckily not always possible, since the flow velocity has a huge spread of possible valus. Thus a mean value for spacing has to be chosen, which is ok for most values. 5.: This unluckily is also not possible. But each "part" is solved separately, thus "inside" each part's PDE the mesh spacing is constant. Just the connection/boundary condition of each PDE with another PDE can have a discontinuity of the mesh spacing. Instabilities are unlikley, since I check for von Neumann stability at each step. $\endgroup$ – Scotty1- Aug 13 '18 at 12:52
  • $\begingroup$ To 6.: But will it still be called a Crank-Nicolson method if I treat the discretization for diffusion with a centered scheme, while using QUICK for the convective term? $\endgroup$ – Scotty1- Aug 13 '18 at 12:54
  • $\begingroup$ To 6: it is fully implicit scheme. $\endgroup$ – ztdep Aug 13 '18 at 13:08
  • $\begingroup$ To 2: QUICK scheme has better convergence character. We can use the absolute stable high resolution schemes, such as MUSCL, MINMOD, etc, also. $\endgroup$ – ztdep Aug 13 '18 at 13:10
  • $\begingroup$ To 2.: Ok, thanks for the information about the other schemes. This looks very interesting and I'll take a look if I can implement it without too much effort. To 6.: Yes, but will the scheme still be called Crank-Nicolson if I change the spatial discretization? Or is the "name" Crank-Nicolson only refering to the trapezoidal rule in time discretization, meaning that I can change the space discretization while still being able to call it Crank-Nicolson? This is just a question about naming conventions, not about the math behind it. :) $\endgroup$ – Scotty1- Aug 13 '18 at 13:17

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