blocku_xx

Information

The u_xx block computes the second-order space derivative. By using the Newton-Gregory backward polynomial we
obtain

where h = x_i+1 - x_i (for i = 0,..., n-1). We assume here that the grid points are spaced equally.
If we wish a second-order central difference approximation, we need to fit the polynomial through the three points
x_i-1, x_i, x_i+1. This means to write the polynomial for example around the point x_i+1 and drop the higher-oder terms to obtain

and finally, to evaluate the second-order space derivative around the point x_i we need to set s = -1 to get

The second-order central difference scheme is implemented in u_xxCD2B2 block.
By following the same approach we can compute the fourth-order central difference scheme. This time we need more terms in the polynomial

and so we obtain

for the boundary points we use a biased formula and we obtain

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