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Steady State Discretisation and Time Step

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Old   August 25, 2021, 09:12
Default Steady State Discretisation and Time Step
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Brett
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Hi everyone,

So I know when solving a steady state problem we use a kind of pseudo time step to march towards convergence. What I want to know is how do we specifically discretise our equations and where is the timestep implemented in this process?
If all the differentials with respect to time can be cancelled due to the problem being steady state where does the time step come in??

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Old   August 25, 2021, 15:36
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There are unsteady/transient solvers that use real time-stepping. If you march these in time, eventually you may arrive at a steady state solution (if it even exists). These always have temporal discretization because they are unsteady solvers and terms cancel out in the solution whenever they feel like canceling out.

There are steady solvers that solve problems that don't depend on time. Steady solvers don't need pseudo time-stepping and most of them don't have any temporal discretization because the terms are cancelled out. But sometimes a steady solver will use a pseudo time-step. Are you asking how these work? The pseudo time-step is a form of implicit underrelaxation and isn't actually a form of a temporal discretization.
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Old   August 25, 2021, 15:43
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Hey LuckyTran.

If you could explain how they work that would be cool!

So theres no temporal discretisation in steady problems?
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Old   August 25, 2021, 16:41
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Alright so for steady problems you have a discretized linear system that needs to be solved. You have an initial guess serving as an old solution. If you solve the linear system you get an updated solution.


With implicit underrelaxation, you can inject (by adding) a certain amount of the old solution into the linear system. When you solve this modified underrelaxed system instead, the updated solution is biased and tends more towards the old solution. This is equivalent to limiting the solution change by a local time-step. You can try and figure out how the weights are related to the local time-step (hint: it's related to the flow Courant number). Or.... you can just put the time-step into the weight factor and that's pseudo transient.
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Old   August 25, 2021, 18:40
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care to indulge me?
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Old   August 25, 2021, 18:46
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For example when I set a physical timescale value in CFX, what exactly am I setting and where does it present itself in the equations (Simplified, say the Euler equations)
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Old   August 26, 2021, 00:05
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The first rule of talking about pseudo transient is that we don't talk about what the real transient PDE is. We start with the presumption and restrict ourselves to only steady pde's. The pseudo transient term looks like so, and you can do this for any steady pde. You just artificially add a transient term into the equation. And any time you inject stuff like this into the linear system, it is a form of implicit under/over-relaxation.



Formally there is a temporal discretization going on, but then you will ask me what is the difference between the transient equation and this steady one with an artificial timestep. Well in this example there isn't a lot. But you should forget that there is a transient form of the N-S equations as long as you are talking about pseudo transient unless you want to confuse yourself. Fluent/CFX uses a 1st order relaxation that looks very much like 1st order backward Euler being applied to a transient PDE. And that's true. But in general, you can artificially add higher order temporal derivatives out of nowhere to further stabilize your solver or accelerate convergence.


The reason the pseudo time-step is useful, is because you can choose the pseudo time-step size based on physical phenomenon that you know exists in your problem to stabilize particular physics (convection, diffusion, buoyancy, etc.) rather than using under-relaxation factors that range somewhere from 0 to 1.
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