Take for instance, a stencil of 3 points can be used for 2nd order approximation of second derivative. Ordinarily, one would need a larger stencil for higher accuracy of the same derivative (second). Now, compact schemes help to avoid larger stencil but maintain higher accuracy. But the downside is that a global linear system is solved for each derivative in the equation.
My points are the following:
(1) Since the linear systems (for derivatives) are global, the entire numerical process is not fully retaining the smaller stencil because every point in the grid is involved in computing the derivative at any other point of the grid. Does this not negate the compactness?
2) These linear systems are obviously as large as the grid (which in practice are very huge), their iterative solution process will be both time and memory consuming. So, putting all these together, do you not think that these issues offset the supposed gain over using larger stencil?
3*) I understand that these schemes are very useful, especially for two important issues, namely (i) avoiding ghost boundary points, and (ii) maintaining a convenient structure for unconditional stability.
I will appreciate if you kindly let me know your thoughts in these.
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