I have heard using CDS scheme can lead to numerical problems due to odd-even decoupling of cells. could you tell me what is the manifestation of this topic in the convergence history or the solution???

Thanks

Hi,

odd-even decoupling of cells occurs when (for example) you use a low (second) order scheme in space and approximate

du/dx = (u(j+1)-u(j-1))/(2*delta x)

Then obviously, by jumping two indexes (from j-1 to j+1; skipping over j), you decouple the even and odd cells in the whole domain (assuming you don't have any other term).

Say, it is a first order equation, therefore, you need one boundary condition (mathematically speaking).

Next, as you impose a boundary condition on one side you actually solve two problems:

1) you solve for the even cells with one boundary condition on one side

2) you solve for the odd cells without any boundary condition (since you impose here only one boundary condition).

This problem will 'manifest' (will show) by two point oscillations in the solution, which are actually two different solution of the same problem (since for each different boundary condition you actually define - mathematically a new problem).

To overcome this problem, you might use another scheme where you do not skip over 'j'; or you might want to use an additional boundary condition, such that the odd and even cells will be coupled (for example if you give the function on the odd cells at one boundary, you might want to consider giving the derivative of the function at the same boundary for the even cells, such that the expression for the derivative includes also the function given at the odd cell boundary - this way you will have a relation between odd and even cell points). Mathematically speaking, this is OK, since the difference equations are decoupled, this means that you have actually two equations of the first order in space and need one boundary condition for each of them.

Patrick Godon

Chees, thanks for your comments!!!