[yhy20081016]

In many CFD codes, spatial discretization scheme is often less accurate than temporal discretization scheme. For example, one might adopt a second-order spatial scheme and use the fourth-order Runge-Kutta method to perform temporal integration. I just don’t understand the reason behind this scenario. Since the spatial scheme is second-order accurate, it seems that a fourth-order accurate temporal scheme doesn’t help to improve the overall accuracy of the solution. Is my point of view correct? Thank you very much.

[FMDenaro]

Originally Posted by yhy20081016

In many CFD codes, spatial discretization scheme is often less accurate than temporal discretization scheme. For example, one might adopt a second-order spatial scheme and use the fourth-order Runge-Kutta method to perform temporal integration. I just don’t understand the reason behind this scenario. Since the spatial scheme is second-order accurate, it seems that a fourth-order accurate temporal scheme doesn’t help to improve the overall accuracy of the solution. Is my point of view correct? Thank you very much.

In principle, you are correct. However, high order time integration is expecially suited for simulation of turbulence, with LES/DNS. This is due to the fact that one wants to exclude effects of filtering/truncation errors over the temporal scales and focus only on spatial scale.
Furthermore, it is quite obvious that for stability reason O(dt) = (1/U) O(h)

[yhy20081016]

Thank you very much.

[duri]

Originally Posted by yhy20081016

In many CFD codes, spatial discretization scheme is often less accurate than temporal discretization scheme. For example, one might adopt a second-order spatial scheme and use the fourth-order Runge-Kutta method to perform temporal integration. I just don’t understand the reason behind this scenario. Since the spatial scheme is second-order accurate, it seems that a fourth-order accurate temporal scheme doesn’t help to improve the overall accuracy of the solution. Is my point of view correct? Thank you very much.

In most of the codes RK is used for multistaging technique. It is not necessarily fourth order accurate. Accuracy depends implementation scheme. Higher order temporal scheme require huge storage space. It is usual to use multistage with low order temporal discretization. The purpose of RK is to damp high frequency errors.

[FMDenaro]

Originally Posted by duri

In most of the codes RK is used for multistaging technique. It is not necessarily fourth order accurate. Accuracy depends implementation scheme. Higher order temporal scheme require huge storage space. It is usual to use multistage with low order temporal discretization. The purpose of RK is to damp high frequency errors.

well, I am not sure what you want to say with "The purpose of RK is to damp high frequency errors" ...
When the dt is fixed, you implicitly cut-off all the frequencies over pi/dt. The frequencies below have to be resolved at best without damping...