On a Class of Additive Iterative Methods

Abstract

In the solution of complicated problems of mathematical physics by numerical methods, great attention is paid to the construction and investigation of efficient algorithms [1–4]. These methods are based on an additive representation of the space operator as the sum of simpler operators, which permits one to pass from a complicated problem to a chain of elementary problems, which admit either sequential or parallel solution. Such methods include classical alternating direction methods, locally one-dimensional finite-difference schemes, and others. Decomposition schemes with respect to separate directions in multidimensional problems, subdomain decomposition schemes [4, 5], and decomposition into physical processes can also be viewed as additive methods. In view of the properties of additive schemes, they occupy an intermediate position between explicit and implicit finite-difference schemes: the amount of computations is close to that of explicit schemes, and from the viewpoint of the stability, they are close to absolutely stable implicit methods.