Finite difference formulae in multi-dimension

TL;DRAbstract

Finite-difference formulae for partial derivatives of functions are essentially one-dimensional formulae. That is, they involve function at grid points along the direction of the derivatives. The one-dimensonal formulae have been employed while tackling the numerical problems in multi-dimension, such as in the numerical solution of partial differential equations. It would be desirable to employ multi-dimensional finite-difference formulae when dealing with problems in multi-dimension. By multi-dimensional finite difference formulae we mean finite-difference formulae involving various points of the multi-dimensional stencil, than just the points of the uni-directional stencil. Use of such finite-difference formulae for numerical problems in multi-dimension is only natural.

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