A Three-Stage Multiderivative Explicit Runge-Kutta Method (original) (raw)
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ON THE CONVERGENCE AND STABILITY OF 2-STAGE MULTIDERIVATIVE EXPLICIT RUNGE–KUTTA METHODS
In an earlier work by the authors [1, 2], a new family of numerical methods for solving Initial Value Problems was proposed and these methods are of higher order than the classical Explicit Runge-Kutta Methods with the same number of internal stages evaluation. In this paper, we examine the convergence and stability properties of these new methods, which are called 2-stage Multiderivative Explicit Runge-Kutta (MERK) methods. The methods are found to be convergent to order 3 and order 4; and they possess wider region of absolute stability than the well-known explicit Runge-Kutta methods with the same internal stages evaluation.
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In this paper, a three-stage fifth-order Runge-Kutta method for the integration of a special third-order ordinary differential equation (ODE) is constructed. The zero stability of the method is proven. The numerical study of a third-order ODE arising in thin film flow of viscous fluid in physics is discussed. The mathematical model of thin film flow has been solved using a new method and numerical comparisons are made when the same problem is reduced to a first-order system of equations which are solved using the existing Runge-Kutta methods. Numerical results have clearly shown the advantage and the efficiency of the new method.
Nine-stage multi-derivative Runge-Kutta method of order 12
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A nine-stage multi-derivative Runge-Kutta method of order 12, called HBT(12)9, is constructed for solving nonstiff systems of first-order differential equations of the form y = f (x, y), y(x 0 ) = y 0 . The method uses y and higher derivatives y (2) to y (6) as in Taylor methods and is combined with a 9-stage Runge-Kutta method. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution leads to order conditions which are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The stepsize is controlled by means of the derivatives y (3) to y (6) . The new method has a larger interval of absolute stability than Dormand-Prince's DP(8,7)13M and is superior to DP(8,7)13M and Taylor method of order 12 in solving several problems often used to test high-order ODE solvers on the basis of the number of steps, CPU time, maximum global error of position and energy. Numerical results show the benefits of adding high-order derivatives to Runge-Kutta methods.
In this research paper, we extended the idea of Hybrid Block method at í µí± = 3 through interpolation and collocation approaches to an effectively Sixth Stage Implicit Runge-Kutta method for the solution of initial value problem of first order differential equations. The new approach displays a uniform order 6 schemes and zero stable. The new method demonstrates superiority over its equivalent linear multi-step method (LMM) with some numerical experiments tested.
Design and Analysis of Some Third Order Explicit Almost Runge-Kutta Methods
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In this paper, we propose two new explicit Almost Runge-Kutta (ARK) methods, ARK3 (a three stage third order method, i.e., s = p = 3) and ARK34 (a four-stage third-order method, i.e., s = 4, p = 3), for the numerical solution of initial value problems (IVPs). The methods are derived through the application of order and stability conditions normally associated with Runge-Kutta methods; the derived methods are further tested for consistency and stability, a necessary requirement for convergence of any numerical scheme; they are shown to satisfy the criteria for both consistency and stability; hence their convergence is guaranteed. Numerical experiments carried out further justified the efficiency of the methods.
Study of Runge Kutta Method of Higher orders and Its Application
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Reformulation on Modified Runge-Kutta Third Order Methods for Solving Initial Value Problems
Abstract – Ordinary Differential equation with Initial Value Problems (IVP) frequently arise in many physical problems. Numerical methods are widely used for solving the problems especially in case of numerical simulation. Several numerical methods are available in the literature for solving IVP. Runge-Kutta (which is actually Arithmetic Mean (AM) based method) is one of the best commonly used numerical approaches for solving the IVP. Recently Evans[1] proposed Geometric Mean (GM) based Runge-Kutta third order method and Wazwaz [2] proposed Harmonic Mean (HM) based Runge-Kutta third order method for solving IVP. Also Yanti et al. [3] proposed the linear combination of AM, HM and GM based Runge-Kutta third order method. We extensively perform several experiments on those approaches to find robustness of the approaches. Theoretically as well as experimentally we observe that GM based and Linear combination of AM, GM and HM based approaches are not applicable for all kinds of problems. To overcome some of these drawbacks we propose modified formulas correspond to those AM and linear combination of AM, GM, HM based methods. Experimentally it is shown that the proposed modified methods are most robust and able to solve the IVP efficiently.
A Simplified Derivation and Analysis of Fourth Order Runge Kutta Method
International Journal of Computer Applications, 2010
The derivation of fourth order Runge-Kutta method involves tedious computation of many unknowns and the detailed step by step derivation and analysis can hardly be found in many literatures. Due to the vital role played by the method in the field of computation and applied science/engineering, we simplify and further reduce the complexity of its derivation and analysis by exploring some possibly well-known works and propose a step by step derivation of the method. We have also shown the stability region graphically