2020-07-12T19:31:28Zhttps://ipsj.ixsq.nii.ac.jp/ej/?action=repository_oaipmhoai:ipsj.ixsq.nii.ac.jp:000113512017-01-12T15:00:00Z00581:00651:00663
Two Stage Explicit Runge-Kutta Type Method Using Second and Third DerivativesTwo Stage Explicit Runge-Kutta Type Method Using Second and Third Derivativeseng論文http://id.nii.ac.jp/1001/00011351/Journal Articlehttps://ipsj.ixsq.nii.ac.jp/ej/?action=repository_action_common_download&item_id=11351&item_no=1&attribute_id=1&file_no=1Copyright (c) 2003 by the Information Processing Society of Japan数値計算アルゴリズムThe University of Electro-CommunicationsCurrent address : 3 -22- 11 Hachimanyama Setagaya Tokyo 156-0056Toshinobu, YoshidaHarumi, OnoA two stage explicit Runge-Kutta type method for solving non-stiff initial-value problems of autonomous ordinary differential equations is proposed.The method uses first- to third-order derivatives of the solution in the first stage and second-order pseudo-derivatives in the second stage;which are the product of the Jacobian matrix of the equations and a vector which is the linear combination of the first-order derivatives and all values obtained in the first stage.In these stages the derivatives and the pseudo-derivatives are assumed to be computed using automatic differentiation.Consequently these computations can be performed quite easily and efficiently.The order conditions of the method are solved and the parameters in the method are shown as functions of a free parameter.This is followed by the presentation of the D2RK245 formulas the fifth-order formula and the fourth-order formula which is embedded in the fifth-order formula.The leading truncation error terms of these formulas as functions of the free parameter are discussed.Finally numerical examples are presented to compare the accuracy CPU time and step control of the proposed method with conventional methods.A two stage explicit Runge-Kutta type method for solving non-stiff initial-value problems of autonomous ordinary differential equations is proposed.The method uses first- to third-order derivatives of the solution in the first stage,and second-order pseudo-derivatives in the second stage;which are the product of the Jacobian matrix of the equations and a vector which is the linear combination of the first-order derivatives and all values obtained in the first stage.In these stages, the derivatives and the pseudo-derivatives are assumed to be computed using automatic differentiation.Consequently, these computations can be performed quite easily and efficiently.The order conditions of the method are solved,and the parameters in the method are shown as functions of a free parameter.This is followed by the presentation of the D2RK245 formulas,the fifth-order formula,and the fourth-order formula which is embedded in the fifth-order formula.The leading truncation error terms of these formulas as functions of the free parameter are discussed.Finally,numerical examples are presented to compare the accuracy, CPU time and step control of the proposed method with conventional methods.AN00116647情報処理学会論文誌44182872003-01-151882-77642009-06-29