An analysis of alternating‐direction methods for parabolic equations

Alternating‐direction solution procedures for parabolic partial differential equations can be developed using finite‐difference, finite‐element, and collocation approximations in space. Each of these methods derives from a common alternating‐direction formulation. Furthermore, each method leads to an O[(Δt)²] error which is in addition to the discretization error associated with standard multidimensional solutions. However, when dealing with equations having spatially varying coefficients, some alternating‐direction formulations lead to yet other errors which are O(Δt). These latter errors, and thus the accuracy of the method, depend on the structure of the mass matrix associated with the approximating method.