A Class Of Multistep Methods For Stiff Ordinary Differential Equations

We often come with stiff differential equations in many fields, such as Control System, Electronic Networks, Biology, Physics, Chemical Dynamics, Aerospace Industry Design and Continuous System Simulation. It is not fit for the explicit methods to solve the stiff problems, which is a significant difference between stiff differential equation and ordinary differential equation. So it requires the implicit methods to solve the stiff problems, which have greater stability region. For nearly three decades, the issue of algorithm theory for rigid problems has aroused widespread concern in many scholars, especially the construction of high-efficient numerical algorithm.In this paper, firstly, we introduced the characteristics of the rigid and some important algorithms, which are including the famous Backward Differentiation Formula(BDF) and Extended Backward Differentiation Formula(EBDF). On this base, we introduced a new class of algorithms, which is the generalization of Extended Backward Differentiation Formula. We deduced the order conditions by analyzing the local errors, and constructed a class of methods which have two parameters.Then, under the premise of satisfying the stability conditions, and through the selection of the parameters, we seek for the method whose stability region is large as possible as it can. And then by using compute to search for the methods, we discussed the methods of k = 4,5,6,7,8 separately. To compare with EBDF, we gave the absolute stability regions. The results showed that the absolute stability regions of the new methods are greater than that of EBDF. Finally, we gave numerical tests about the one-dimensional and two-dimensional of stiff differential equations. The tests'results were accord with the results of the theoretical analysis.