The Study Of Some Linear Multi-step Methods With The More Bigger Absolutely Stable Region

In the control system, biology, physics, electronic networks, aerospace industry, chemic-al dynamics, design and continuous system simulation, we often come with the stiff different-ial equations. It is fit for the implicit methods to solve the stiff problems, which is different w-ith ordinary differential equation. So it requires the implicit methods with greater stable regio-n. For nearly years. Many scholars have interested with the issue of algorithm theory for stiff problems, and studyed it, especially the construction of high-efficient numerical algorithm.Linear multi-step methods and Runge-Kutta law are the typical and high-efficient nume-rical algorithm to solve initial value problems of ordinary differential equation. Forms of line-ar multi-step methods is simple and easy to compute. So it is one of the more comprehensive methods. The Runge-Kutta method is a type of the important classic algorithm that solves the stiff differential equation. This text study more larger absolutely stable region for linear multi-step methods. We know a kind of good computing method should be small computing quanti-ty and satisfied a computing result that specify accuracy's request. Therefore the error is the important beacon for computing method good or bad. Only considering theory error, namely local error, isn't enough, have to also computing error. The consistency decides the size of lo-cal discrete error, but zero stability mean spreading of error. But when using smaller step-len-gth and bigger N under of [a, b], If like that, it means not only computing time heavybut also the accuracy of the error badly. This explains the scarcity of zero stability, for getting given accuracy, finding out suitable computing step length, introducing the notion of absolute stab-ility, the stability even has more meaning. Dimension and shape of the absolute stability reg-ions is important to discover to a kind of valuable method. The more larger absolute stability region,the smaller restriction to step length for the specific eigenvalue, Specially stiff differe-ntial equations, the eigenvalue of system is more bigger than absolutely stable interval, step length will be limited. So we hope that the absolutely stable region is large as possible as it can.In this paper, firstly, we introduced the characteristics of the stiff and some important algorithms. which are including the famous linear multi-step methods that Gear construct k-order and Runge-Kutta method. On this base. we studyed a new class of algorithms. which is the linear multi-step methods. In the one dimensional regions, to analyze absolutely stable re- gion. Namely, analyzing by the condition that absolutely stable region intersects with negative axis. We introduced a class method of linear two-step of three-order, which have one parame-ter, a class method of linear three-step of four-order, which have two parameters, and a class method of three-step of three order which is the generalization of Adams-Moulton and have one parameter. Then, under the condition of satisfying the stability, and through the selection of the parameters, we seek for the method whose stability region is large as possible as it can. To draw their absolutely stable region by margin track law, comparing with absolutely stable region of Adams-Moulton, The results showed that the absolutely stable region of the new m-ethods are greater than that of Adams-Moulton, and better for stiff differential equation. Com-paring with Gear, it have no larger absolutely stable region, but higher accuracy. and is still meaningful.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.