[jakubstary]

What is main advantage of using 2nd and 4th order Runge Kutta methods for time discretization? Is it stability for larger CFL condition or Runge Kutta is more accurate for same CFL condition?

I compute Shallow water equations and I use Euler method, RK2 and RK4 for time discretization. If I have CFL = 0.9 then I get Euler method as the most accurate, but if I set CFL = 3.5 then I get RK4 is more accurate than Euler method with CFL=0.9. Euler method is obviously unstable for CFL = 3.5.

Are my results correct?

Thank you
Jakub

[jakubstary]

I have tested the Euler method and the RK2 method for increasing CFL and I get this result. Don't anyone know if it's correct result? If not, don't you have an idea where might be error in my calculations?

Thank you.

[Aeronautics El. K.]

Euler method is first order accurate while RK4 is forth order accurate. Moreover, the explicit Euler method has quite strict stability criteria.

[jakubstary]

And is it correct, that Euler is more accurate for small CFL condition than RK2 and RK4? I thought RK2 is always more accurate than Euler, but in my program not, see graph.

Thank you very much.
Jakub

[Aeronautics El. K.]

e is the temporal discretisation error?

[jakubstary]

e is sum L1 error between numerical solution and exact solution of Riemann problem.

[Aeronautics El. K.]

what problem are you solving? NS or Euler Equations? How is the spatial discretisation done?

[jakubstary]

I'm solving Euler equations. For case which is on graph I used simple Lax-Friedrichs scheme for spatial discretisation.

[jakubstary]

Please, Aeronautics El. K. don't you have any idea, why Runge Kutta behave strangely for my case?

Thank you.

[Aeronautics El. K.]

I can't make anything of it yet. I'm reading a little bit on it and I suggest you do the same

[jakubstary]

Aeronautics El. K.: Of course I do the same :-) I'm trying to solve it for about two weeks :-(. Thank you for your willingness to help.

If someone will have some idea or tip for book, where can I find it, please write it here. Many thanks.

Jakub

[jakubstary]

I discovered an interesting thing. I found Masatsuka's code http://www.cfdbooks.com/cfdcodes/oned_euler_v1.f90, where he solve same problem as me. I implemented even Euler method to his code. I compared results obtained using RK2 and Euler method for time discretization and Euler is more accurate. So really advantage of using RK2 and RK4 instead of Euler is probably only possibility using larger timestep (or CFL)? I think that RK2 and RK4 may not be always more accurate.

I don't if this is correct conclusion of my problem :-). Books say something else, but numerical results not.

Jakub

[emjay]

Hello,

this thread is almost 7 years old, but i think it is important knowing the real reason why for hyperbolic PDEs a higher order time integration could be less accurate.

In hyperbolic PDEs solution with discontinuities are possible.
Every spatial discretisation higher than first order produce non-physical solution.
see Godunov's theorem

To ensure Total Variation Diminishing property TVD it is also necessary having a time discretisation which don't violate the TVD property.

For time integration Euler Method (explicit) ensure the TVD Property, and
also higher order Runge Kutta TVD schemes (RK2-TVD and RK3-TVD) are suitable for this class of equations.

Literatur: S. Gottlieb, C.W. Shu, Total variation diminishing Runge-Kutta schemes, Math. Comput. 67 (1998) 73–85.

[FMDenaro]

Originally Posted by emjay

In hyperbolic PDEs solution with discontinuities are possible.
Every spatial discretisation higher than first order produce non-physical solution.
see Godunov's theorem

Just to address that the Godunov's theorem stated that only linear first order accurate schemes are monotone.

You can have physical solution using higher order non-linear scheme.

[emjay]

Originally Posted by FMDenaro

Just to address that the Godunov's theorem stated that only linear first order accurate schemes are monotone.

You can have physical solution using higher order non-linear scheme.

yes that is true, but the flux is "modified" to ensure bounded values.

But to preserve monotonicity it is also necessary having a TVD scheme as temproral Discretization.

And i think that the thread creator have/had this problems with higher order time integration schemes.

https://www.researchgate.net/publica...-Kutta_Schemes