Here, we present a new algorithm that systematically solves the Taylor expansion coefficients problem for constructing implicit (compact) finite-difference schemes.
Although it is presented to construct up to 3rd-order differential compact schemes, we believe it is simple enough, so that users can easily extend it to obtain even higher-order schemes if necessary.

Also, we provide two examples:
The first example, demonstrates how to use the Taylor Table algorithm to recover well-known schemes in the literature. The second example, shows how to set a central compact scheme and complement it with suitable boundaries schemes. So that a (sparse) differential operator can be easily constructed ;)

Future work: In an expansion of these snippets I'll soon introduce a simple way to create 2D and 3D differential operators suitable for solving PDEs in Matlab ~stay tuned !

-M. Diaz

Happy coding !