A unified approach to high-order compact finite difference schemes for 3D Poisson equations

A unified approach to high-order compact finite difference schemes for solving three-dimensional Poisson equations is derived. Notice that all high order compact finite difference schemes are linear combinations of the solution values at grid points and source terms at selected points. In the unified strategy, a parameter γ is introduced for the linear combination of the solution values at the compact finite difference stencil. In our approach, carefully chosen γ ’s lead to different fourth-order schemes. Carefully chosen linear combination of the source term at grid points will lead to different fourth-order schemes. If additional points such as the middle points are used, sixth-order schemes can be achieved if the solutions with certain symmetries. The convergence proof is provided for the new unified scheme based on the M -matrix condition along with non-trivial numerical examples.