Expansion of solutions of parameterized equations and acceleration of numerical methods

A general scheme of parameterized families of equations is considered, and abstract results on the expansion of the solutions and on the acceleration of their convergence in terms of the parameter are presented. These results are applied to fractional step approximations for linear parabolic PDEs, systems of linear PDEs, and for nonlinear ordinary differential equations. Applications to accelerating the convergence of finite difference schemes for these equations will be presented in a subsequent paper.