The aim of this research is to study the efficiency of symplectic and non-symplectic Runge-Kutta methods in solving Kepler problem. The numerical behavior of the Runge-Kutta (RK) methods that are symmetric such as the implicit midpoint rule (IMR), implicit trapezoidal rule (ITR), 2-stage and 2-stage Gauss (G2) method are compared with the non-symmetric Runge-Kutta methods such as the explicit and implicit Euler (EE and IE), explicit midpoint rule (EIMR), explicit trapezoidal rules (EITR), explicit 4-stage Runge-Kutta (RK4) method and 2-stage Radau IIA method (R2A). Kepler problem is one type of nonlinear Hamiltonian problem that describes the motion in a plane of a material point that is attracted towards the origin with a force inversely proportional to the distance squared. The exact solutions phase diagram produces a unit circle. The non-symplectic methods only reproduce a unit circle at certain time intervals while the symplectic methods do produce a unit circle at any time interva.....

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The purpose of this research is to investigate the e?ciency of ex-plicit diagonally implicit multi-stage integration methods with ex-trapolation. The author gave detailed explanation of explicit di-agonally implicit multi-stage integration method and compared thebase method with a technique known as extrapolation to improvethe e?ciency. Extrapolation for symmetric Runge-Kutta method isproven to improve the accuracy since with extrapolation the solu-tions exhibit asymptotic error expansion, however for General linearmethods, it is not known whether extrapolation can improve the ef-?ciency or not. Therefore this research focuses on the numericalexperimental results of the explicit diagonally implicit multistageintegration with and without extrapolation for solving some ordi-nary di?erential equations. The numerical results showed that thebase method with extrapolation is more e?cient than the methodwithout extrapolation  ..

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The extrapolation technique has been proved to be very powerful in improving the performance of the approximate methods by large time whether engineering or scientific problems that are handled on computers. In this paper, we investigate the efficiency of extrapolation of explicit general linear methods with Inherent Runge-Kutta stability in solving the non-stiff problems. The numerical experiments are shown for Van der Pol and Brusselator test problems to determine the efficiency of the explicit general linear methods with extrapolation technique. The numerical results showed that method with extrapolation is efficient than method without extrapolation...

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The main aim of this article is to present recent results concerning diagonal implicitmultistage integration methods (DIMSIMs) with extrapolation in solving stiff problems. Implicit methods with extrapolation have been proven to be very useful in solving problems with stiff components. There are many articles written on extrapolation of Runge-Kutta methods however fewer articles on extrapolation were written for general linear methods. Passive extrapolation is more stable than active extrapolation as proven in many literature when solving stiff problems by the Runge-Kutta methods. This article takes the first step by investigating the performance of passive extrapolation for DIMSIMs type-2 methods. In the variable stepsize and order codes, order-2 and order-3 DIMSIMs with extrapolation are investigated for Van der Pol and HIRES problems. Comparisons are made with ode23 solver and the numerical experiments showed that implicit DIMSIMs with extrapolation has greater accuracy than the met.....

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Diagonally implicit multistage integration methods (DIMSIMs) are widely utilized in finding the solution to any problems in the subject of ordinary differential equations. These methods are selected from the general linear methods, which is considerable potential for efficient implementations. The extrapolation is derived from the stability of the explicit Runge-Kutta methods. In this paper, the combination of DIMSIMs with Richardson extrapolation of different orders shows that numerical solutions give higher accuracy when the extrapolation is applied with the base method. 2023 Penerbit UTM Press...

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