[cfduser999]

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 steady-state Euler equation (both 1d) to know the steady-state solution.

After fullfiting the CFL condition, with the Courant number ~0.33, I can well reproduce the steady-state solution, but if I keep reducing the time-stepping, I will reach another steay-state solution, which also looks more weird. This confuses me profundly, cause I think smaller time-stepping should produce the same steady-state solution, just with higher resolution. Can anyone help?

[FMDenaro]

Originally Posted by cfduser999

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 steady-state Euler equation (both 1d) to know the steady-state solution.

After fullfiting the CFL condition, with the Courant number ~0.33, I can well reproduce the steady-state solution, but if I keep reducing the time-stepping, I will reach another steay-state solution, which also looks more weird. This confuses me profundly, cause I think smaller time-stepping should produce the same steady-state solution, just with higher resolution. Can anyone help?

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.

[cfduser999]

Originally Posted by FMDenaro

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.

Thank you for your reply. It's just a simple gas flow through a cylinder tube with known boundary conditions, and is fully explicit.

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

[cfduser999]

Originally Posted by FMDenaro

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.

And can you please point me to some references regarding that "reducing only the time step could highlight the remaining part of the local truncation error due to the spatial discretization"? Thank you in advance.

[FMDenaro]

Originally Posted by cfduser999

And can you please point me to some references regarding that "reducing only the time step could highlight the remaining part of the local truncation error due to the spatial discretization"? Thank you in advance.

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.

[cfduser999]

Originally Posted by FMDenaro

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.

Thank you for your reply.

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

[LuckyTran]

Originally Posted by cfduser999

And can you please point me to some references regarding that "reducing only the time step could highlight the remaining part of the local truncation error due to the spatial discretization"? Thank you in advance.

You can prove it very quickly

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.

[FMDenaro]

Originally Posted by cfduser999

Thank you for your reply.

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

I see, there are several textbook where you can study the modified equation, for example Anderson or Hirsch

[cfduser999]

Originally Posted by LuckyTran

You can prove it very quickly

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.

I understood, reducing the time step without reducing the spatial step increases essentially the global error. Thank you very much!

[FMDenaro]

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*(1-c)*d^2f/dx^2 + ...


you can see what happens when c is reduced and dx is fixed.

[cfduser999]

Originally Posted by FMDenaro

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*(1-c)*d^2f/dx^2 + ...


you can see what happens when c is reduced and dx is fixed.

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