An efficient optimized adaptive step-size hybrid block method for integrating differential systems

Singh, Gurjinder; Garg, Arvind; Kanwar, V.; Ramos, Higinio

doi:10.1016/j.amc.2019.124567

Cited by 23 publications

(21 citation statements)

References 36 publications

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“…We compared certain novel subclasses to implicit RKtype methods, i.e., the established collocation methods with the same framework and purpose, and we demonstrated the advantage of the new methods. Certain extensions and alternatives could outperform the proposed approach: the embedded formula collocation method [20,25], multivalue methods [26], parallel time integration methods [27], variable step size methods [28], block methods [29][30][31][32], etc. We focused on the popular RK-type methods to avoid an extensive comparison to all the possible alternatives.…”

Section: Discussionmentioning

confidence: 99%

Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Urevc

Halilovic̆

2021

Mathematics

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In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

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Section: Discussionmentioning

confidence: 99%

Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Urevc

Halilovic̆

2021

Mathematics

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“…Note that these formulas are in terms of the intra-step points x n+u , x n+r , x n+s , x n+t . In order to find appropriate values of these intra-step points, expanding the formulas at x n+s and x n+1 about Taylor series about the point x n , we get (y(x n+s ), h) = s 3 (A + sB) y (7)…”

Section: Development Of a Two-parameter Class Of Methodsmentioning

confidence: 99%

“…In the literature, many researchers proposed the methods having the combination of hybrid and block methods. 6,7,[11][12][13]17,18,20,[31][32][33][34][35] Many authors proposed the hybrid methods with the arbitrary values to the off-step points (e.g., one can see , then the Ramos et al 35 first proposed the concept of finding the values to the off-step points, which improves the efficiency of the method in terms of accuracy.…”

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confidence: 99%

A novel two‐parameter class of optimized hybrid block methods for integrating differential systems numerically

Singh

Garg²,

Singla

et al. 2021

Comp and Math Methods

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In this article, a two-parameter class of hybrid block methods for integrating first-order initial value ordinary differential systems is proposed. The methods exhibit hybrid nature which helps in bypassing the first Dahlquist barrier existing for linear multistep methods. The approach used in the development of a class of methods is purely interpolation and collocation technique. The class of methods is based on four intra-step points from which two intra-step points have been optimized by using an optimization strategy. In this optimization strategy, the values of two intra-step points are obtained by minimizing the local truncation errors of the formulas at the points x n+1∕2 and x n+1 .The order of accuracy of the proposed methods is six. A method as a special case of this class of methods is considered and developed into a block form which produces approximate numerical solutions at several points simultaneously. Further, the method is formulated into an adaptive step-size algorithm using an embedded type procedure. This method which is a special case of this class of methods has been tested on six well-known first-order differential systems. K E Y W O R D Sblock methods, embedded-type procedure, hybrid methods, ordinary differential equations INTRODUCTIONNumerous numerical integrators are available for solving first-order initial value ordinary differential systems numerically. Runge-Kutta and linear multistep methods are the usual classes of methods that are being used for solving these systems numerically. These days many built-in computer codes based on these classes of methods are available in software like Matlab, Mathematica, and so forth. To develop more accurate and efficient codes for solving the differential systems is an ongoing process of research. This article is concerned with the development of a class of optimized codes for solving the following system:whereTo proceed further, we assume that the system (1) has a unique continuously differentiable solution that we denote by Y(x). As a first step, the interval of interest [x 0 , x N ] is discretized as follows:x n = x 0 + nh, n = 0, 1, 2, 3, … , N; h = x n+1 − x n and an approximate numerical solution of the system (1) at x n is denoted as Y n ≈ Y(x n ).

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“…Block methods have the advantage over Runge-Kutta methods that they are less expensive in terms of number of function evaluations for a given order. In recent years, work on block methods (Singh et al 2019;Ramos and Singh 2017;Singla et al 2021) has drawn attention of many researchers.…”

Section: Introductionmentioning

confidence: 99%

Using a cubic B-spline method in conjunction with a one-step optimized hybrid block approach to solve nonlinear partial differential equations

2021

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In this paper, we develop an optimized hybrid block method which is combined with a modified cubic B-spline method, for solving non-linear partial differential equations. In particular, it will be applied for solving three well-known problems, namely, the Burgers equation, Buckmaster equation and FitzHugh–Nagumo equation. Most of the developed methods in the literature for non-linear partial differential equations have not focused on optimizing the time step-size and a very small value must be considered to get accurate approximations. The motivation behind the development of this work is to overcome this trade-off up to much extent using a larger time step-size without compromising accuracy. The optimized hybrid block method considered is proved to be A-stable and convergent. Furthermore, the obtained numerical approximations have been compared with exact and numerical solutions available in the literature and found to be adequate. In particular, without using quasilinearization or filtering techniques, the results for small viscosity coefficient for Burgers equation are found to be accurate. We have found that the combination of the two considered methods is computationally efficient for solving non-linear PDEs.

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An efficient optimized adaptive step-size hybrid block method for integrating differential systems

Cited by 23 publications

References 36 publications

Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

A novel two‐parameter class of optimized hybrid block methods for integrating differential systems numerically

Using a cubic B-spline method in conjunction with a one-step optimized hybrid block approach to solve nonlinear partial differential equations

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