when I am approximating the second derivative from the first derivative, shall I change my steps and use interpolations? Or I can use some other ways?
Has there been any research into what a sufficiently small grid size is? I assume this would cause aliasing if you don't get it right.
Are there theorems to determine how fine the grid would need to be to avoid aliasing?
How does the error introduced in the first approximation propagate to the second approximation?
Will the approximation errors compound?
How to extend this approximation to 2-D or even more dimensions? Is there any additional thing we need to do on the mesh?
Is there any example of other ways to discretize the second derivative?
Seconding the question on if approximation errors compound, especially since the wave equations were mentioned to have a similar sort of property with errors.
When computing both the first and second derivative, do we always care about the h/h%2 relationship or do we only care that they are sufficiently small numbers?
How do we avoid the compounding of errors? Since this method involves approximation.
Why doesn't it work well on other types of grids?
How much error is in this approximation?
What are some edge cases where the approximations would produce huge error?
How does the error play out in this method of approximating, as we are approximating twice would the errors compound?