[jones123]

Hello all,

I'm trying to get a better understanding of what the Courant number actually means. Therefore, I am trying to visualize it schematically.

I know that, for example when using a first order upstream scheme, the information flows like this (see attachmant) when the Courant number is between 0 and 1. In this image, n = time and j = position.

But how do these flows change when the Courant number changes? For example, I can't seem to figure out how to draw such a scheme with information flows when the Courant number is like -1, +2 or +3 or whatever.

Can someone help me? (drawing these schemes would be very nice).

Thanks in advance!

[FMDenaro]

Originally Posted by jones123

Hello all,

I'm trying to get a better understanding of what the Courant number actually means. Therefore, I am trying to visualize it schematically.

I know that, for example when using the upstream scheme, the information flows like this (see attachmant) when the Courant number is between 0 and 1. In this image, n = time and j = position.

But how do these flows change when the Courant number changes? For example, I can't seem to figure out how to draw such a scheme with information flows when the Courant number is like -1, +2 or +3 or whatever.

Can someone help me? (drawing these schemes would be very nice).

Thanks in advance!

Do not consider negative values as they simply state the flow has negative direction (u<0).

As you consider c>1, it simply states that c=u/(h/dt) = (dx/dt)/(h/dt) > 1.
Geometrically, you have that the slope of the characteristic curve (dx/dt) is greater than the discrete counterpart (h/dt).
In your sketch, the discrete curve has intersection with the axis tn outside (j-1,j). For example c= 2 intersects j-2, c=3 intersects j=3 and so on.
In a first order upwind, you can intepret that fact as an extrapolation of the vlaues, which actually drives to a numerical instability.

[jones123]

Originally Posted by FMDenaro

Do not consider negative values as they simply state the flow has negative direction (u<0).

As you consider c>1, it simply states that c=u/(h/dt) = (dx/dt)/(h/dt) > 1.
Geometrically, you have that the slope of the characteristic curve (dx/dt) is greater than the discrete counterpart (h/dt).
In your sketch, the discrete curve has intersection with the axis tn outside (j-1,j). For example c= 2 intersects j-2, c=3 intersects j=3 and so on.
In a first order upwind, you can intepret that fact as an extrapolation of the vlaues, which actually drives to a numerical instability.

Hi,

Thanks for the quick reply. Does it then mean something like this?

Thanks

[FMDenaro]

yes, the green lines are the slopes of h/dt for increasing c number ...you see that they lie outside (j-1,j)