Complete centered finite difference method for Helmholtz equation.

A new approach in the finite difference framework is developed, which consists of three steps: choosing the dimension of the local approximation subspace, constructing a vector basis for this subspace, and determining the coefficients of the linear combination. New schemes were developed to form the basis of the local approximation subspace, which were derived by approximating only the k 2 u term of the Helmholtz equation. The construction of a basis of the local approximation subspace allows the new approach to be able to represent any finite difference scheme that belongs to this subspace.