Additive methods for the numerical solution of ordinary differential equations

Consider a system of differential equations $x\prime = f(x)$. Most methods for the numerical solution of such a system may be characterized by a pair of matrices (A, B) and make no special use of any structure inherent in the system. In this article, methods which are characterized by a triple of matrices $(A;{B_1},{B_2})$ are considered. These methods are applied in an additive fashion to a decomposition $f = {f_1} + {f_2}$ and some methods have pronounced advantages when one term of the decomposition is linear. This article obtains algebraic conditions which give the order of convergence of such methods. Some simple examples are displayed.