Implicit Linear Multistep Methods with Nonnegative Coefficients for Solving Initial Value Problems

概要

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Consider the linear multistep methods (LM methods)y_<n+k>+a_&k-1>y_<n;k-1>+...+a_0y_n=h(β_kf_<n+k>+...+β_0f_n) for solving initial value problems of ordinary differential equations. In many of the methods, the signs of α's andβ's are mixed. However, the methods satisfying the conditions -α_j≧0,1,...,k-1,β_j0, j=0,1,...,k are preferable to others because these conditions prev.ent the cancellation of significant figures during the computations. In this paper, we consider the existence of the implicit LM methods satisfying these conditions, for each of the three types; these types consist of the Adams type, the Milne type, and the Radial type. It is found that the highest orders of such LM methods are 2, 4, and 8, for the Adams, for the Milne, and for the Radial, respectively. In particular, the Adams type includes the A_o-and A-stable methods. For the Milne type, it is found that the methods of order 3, 4 are unstable, and consequently only the one of order 2 is useful. For the Radial type of order from 3 to 5, the optimal parameters are obtained which minimize the round-off error propagations of the methods. The numerical example shows that these optimal methods are more accurate than the Adams-Moulton methods.
一般社団法人情報処理学会の論文
1989-03-31