Advances in the theory of variable stepsize variable formula methods for ordinary differential equations

摘要

The theory of the constant stepsize constant formula methods (CSCFM's) for solving initial value problems for ordinary differential equations (ODE's) is well-developed. However, the integration methods are often used as variable stepsize variable formula methods (VSVFM's) in the codes for automatic solution of ODE's. This fact explains why it is necessary to develop a rigorous theory for the VSVFM's also.Results obtained until now in the efforts to develop a theory for the VSVFM's are presented in a systematic way and compared with the well-known results for CSCFM's. The comparison is used to outline the directions in which further efforts in the attempt to improve the theoretical results and to obtain new results are most needed.The integration methods can be separated into two large groups: implicit methods and explicit methods (including here the predictor-corrector schemes). It is discussed what kind of results are needed for each of these two groups.Assume that a set F of CSCFM's is given. From a practical point of view it is highly desirable that the CSCFM's belonging to set F have the following property: if every CSCFM from set F is consistent, zero-stable and convergent, then any VSVFM induced by the CSCFM's of set F is also consistent, zero-stable and convergent for a sufficiently large class of grids. A theorem, concerning a class of CSCFM's with this property, is presented. The application of this theorem in the selection of methods for solving large systems of ODE's arising after the space discretization of air pollution models is discussed. It is explained why the VSVFM's based on this class provide a very efficient computational process in the treatment of air pollution models.