Study On Linear Multistep Method Basic Formulas Of Initial Value Problems In Ordinary Differential Equations

In this paper, the linear multistep formulas of the initial-value problem in ordinary differential equations are further researched on the base of the present situation. The basic formulas of the linear multistep method are defined. Then deduced all linear 2-3 step method basic formulas and chose out convergent formulas from them. The orders, error and stability of these convergence formulas are discussed. And a new class of formulas is given.The major tasks in this paper include:Firstly, the basic formulas of the linear multistep method are defined. We called a k-step formula basic formula if its order is not less than k.Secondly, on the base of the structure theorem of the multistep method, we deduced all 2-3 step basic formulas using symbol operation of Matlab.Then, by means of the convergent condition of the linear multistep method, we attained the all convergent formulas of linear 2-3 step method and calculated the coefficients of the formulas, errors coefficients, orders and intervals of absolute stability.Furthermore, we drew the figures of region of absolute stability of the formulas that are absolutely stable by means of root locus method and analyzed the stability of these formulas.Finally, a new class of linear k-step implicit methods of order k, similar to Gear methods, is given. The fractal coefficients of the formulas of order 2 to 6 in this class are deduced, the orders and error coefficients are also gotten. Then we proved that they are A(α)-stable and the radiant angles of stabilityαare given. In the end, by means of the comparison numerical experiment, it is verified that they are stable and effective for solving initial-value problems of stiff ordinary differential equations.The results of this paper have marked significance in both theoretical research and application of the numerical method in the initial-value problem in ordinary differential equations.