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Euler equation reaches different steadystate solution with different time stepping 

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July 9, 2024, 11:40 
Euler equation reaches different steadystate solution with different time stepping

#1 
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Hi everyone,
I'm currently solving the Euler equation with the Lax method with matlab to see the transient behaviour of my gas parameters. Previous to this I have solved the steadystate Euler equation (both 1d) to know the steadystate solution. After fullfiting the CFL condition, with the Courant number ~0.33, I can well reproduce the steadystate solution, but if I keep reducing the timestepping, I will reach another steaystate solution, which also looks more weird. This confuses me profundly, cause I think smaller timestepping should produce the same steadystate solution, just with higher resolution. Can anyone help? 

July 9, 2024, 12:54 

#2  
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Filippo Maria Denaro
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Could you better detail your flow problem? Is your scheme fully explicit? In principle, the solution should show convergence at constant CFL, that is reducing both the time ans the space step. Reducing only the the time step could highlight the remaining part of the local truncation error due to the spatial discretization. Post the figures of your solution. 

July 10, 2024, 03:16 

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Then does it mean the solution would change with different Courant number? In that case, how do I know which is the right solution? 

July 10, 2024, 03:20 

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July 10, 2024, 04:50 

#5 
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Filippo Maria Denaro
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What you are asking is something that is the basis of any CFD student. What is your background? You will find this topic in any CFD textbook, just study the modified differential equation.


July 10, 2024, 04:53 

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I studied physics but not in the direction of gas/fluid dynamics, I will check some CFD textbook. 

July 10, 2024, 05:00 

#7  
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Lucky
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The Courant number is low with coarse grids. If you have a very coarse grid the Courant number becomes very very low and you fool yourself. Now you have a spatial discretization error that remains even for stupidly small timesteps. This is why we tell people to always do sensitivity studies and validate their tools for each specific new problem. The simplicity of the problem doesn't mean that common errors that plague numerical codes don't exist, actually it makes it even easier to find them. Stable does not necessarily mean accurate. 

July 10, 2024, 05:41 

#8 
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Filippo Maria Denaro
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July 10, 2024, 06:08 

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July 10, 2024, 06:13 

#10 
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Filippo Maria Denaro
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Just as an example the first order upwind for the linear wave equation produces a modified equation like this
df/dt+u*df/dx= 0.5*dx*(1c)*d^2f/dx^2 + ... you can see what happens when c is reduced and dx is fixed. 

July 10, 2024, 06:38 

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Understood, the dissipation would become huge. Thank you very much!


Tags 
cfd, courant number, lax method, time step 
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