A third-order Nakashima pseudo-Runge-Kutta method

In this paper, we present a third order Nakashima pseudo-Runge-Kutta method. This method is derived by minimizing the error bound to determine the free parameters. Since the proposed method is only two-stages, it is cheaper than the traditional method. The stability of the method is analyzed and the numerical results are tabulated to compare with the traditional method

Explicit two-step Runge-Kutta methods

summary:The explicit two-step Runge-Kutta (TSRK) formulas for the numerical solution of ordinary differential equations are analyzed. The order conditions are derived and the construction of such methods based on some simplifying assumptions is described. Order barriers are also presented. It turns out that for order $p\le 5$ the minimal number of stages for explicit TSRK method of order $p$ is equal to the minimal number of stages for explicit Runge-Kutta method of order $p-1$. Numerical results are presented which demonstrate that constant step size TSRK can be both effectively and efficiently used in an Euler equation solver. Furthermore, a comparison with a variable step size formulation shows that in these solvers the variable step size formulation offers no advantages compared to the constant step size implementation

Implementation of two-step Runge-Kutta methods for ordinary differential equations

AbstractWe investigate the potential for efficient implementation of two-step Runge-Kutta methods (TSRK), a new class of methods introduced recently by Jackiewicz and Tracogna for numerical integration of ordinary differential equations. The implementation issues addressed are the local error estimation, changing stepsize using Nordsieck technique and construction of interpolants. The numerical experiments indicate that the constructed error estimates are very reliable in a fixed and variable stepsize environment

Some investigations into the numerical solution of initial value problems in ordinary differential equations

PhD ThesisIn this thesis several topics in the numerical solution of the initial value problem in first-order ordinary diff'erentlal equations are investigated, Consideration is given initially to stiff differential equations and their solution by stiffly-stable linear multistep methods which incorporate second derivative terms. Attempts are made to increase the size of the stability regions for these methods both by particular choices for the third characteristic polynomial and by the use of optimization techniques while investigations are carried out regarding the capabilities of a high order method. Subsequent work is concerned with the development of Runge-Kutta methods which include second-derivative terms and are implicit with respect to y rather than k. Methods of order three and four are proposed which are L-stable. The major part of the thesis is devoted to the establishment of recurrence relations for operators associated with linear multistep methods which are based on a non-polynomial representation of the theoretical solution. A complete set of recurrence relations is developed for both implicit and explicit multistep methods which are based on a representation involving a polynomial part and any number of arbitrary functions. The amount of work involved in obtaining mulc iste, :ne::l'lJds by this technique is considered and criteria are proposed to Jecide when this particular method of derivation should be em~loyed. The thesis is conclud~d by using Prony's method to develop one-step methods and multistep methods which are exponentially adaptive and as such can be useful in obtaining solutions to problems which are exponential in nature

エネルギー関数を持つ発展方程式に対する幾何学的数値計算法

学位の種別: 課程博士審査委員会委員 : (主査)東京大学教授 松尾 宇泰, 東京大学教授 中島 研吾, 東京大学准教授 鈴木 秀幸, 東京大学准教授 長尾 大道, 東京大学准教授 齋藤 宣一University of Tokyo(東京大学