| Popis: |
Numerical methods for solving initial value problems in ordinary differential equations are studied. A notation is introduced to represent cyclic methods in terms of two matrices, Ah, and Bh, and this is developed to cover the very extensive class of m-block methods. Some stability results are obtained and convergence is analysed by means of a new consistency concept, namely optimal consistency. It is shown that optimal consistency allows one to give two-sided bounds on the global error, and examples are given to illustrate this. The form of the inverse of Ah is studied closely to give a criterion for the order of convergence to exceed that of consistency by one. Further convergence results are obtained , the first of which gives the orders of convergence for cases in which Ah, and Bh, have a special form, and the second of which gives rise to the possibility of the order of convergence exceeding that of consistency by two or more at some stages. In addition an alternative proof is given of the superconvergence result for collocation methods. In conclusion the work covered is set in the context of that done in recent years by various authors. |
