An efficient variable step-size rational Falkner-type method for solving the special second-order IVP

Ramos, Higinio; Singh, Gurjinder; Kanwar, V.; Bhatia, Saurabh

doi:10.1016/j.amc.2016.06.033

Cited by 17 publications

(13 citation statements)

References 11 publications

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“…Thus, the development of novel numerical integration methods is of great interest in applied mathematics. In the last decade, many highly efficient solvers were proposed, including Falkner-type block methods [1,2], symmetric multistep methods [3], Störmer-Cowell methods [4], and single-step composition schemes [5]. Great attention is usually paid to the convergence and stability of designed methods, especially in the case of partial differential equations solvers [6].…”

Section: Introductionmentioning

confidence: 99%

“…It has been shown that ESIMM methods outperform classical multistep solvers, such as Adams-Bashforth (AB), Adams-Moulton (AM) and backward differentiation formula (BDF) being implemented with a fixed integration stepsize. However, the most advanced approaches for solving ODEs usually involve adaptive stepsize techniques [1,4]. Conventional local truncation error (LTE) estimators, such as the embedded methods approach [14], appeared to be barely suitable for ESIMM due to its specific method of right-hand side (RHS) calculation.…”

Section: Introductionmentioning

confidence: 99%

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Adaptive Stepsize Control for Extrapolation Semi-Implicit Multistep ODE Solvers

Butusov

2021

Mathematics

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Developing new and efficient numerical integration techniques is of great importance in applied mathematics and computer science. Among the variety of available methods, multistep ODE solvers are broadly used in simulation software. Recently, semi-implicit integration proved to be an efficient compromise between implicit and explicit ODE solvers, and multiple high-performance semi-implicit methods were proposed. However, the computational efficiency of any ODE solver can be significantly increased through the introduction of an adaptive integration stepsize, but it requires the estimation of local truncation error. It is known that recently proposed extrapolation semi-implicit multistep methods (ESIMM) cannot operate with existing local truncation error (LTE) estimators, e.g., embedded methods approach, due to their specific right-hand side calculation algorithm. In this paper, we propose two different techniques for local truncation error estimation and study the performance of ESIMM methods with adaptive stepsize control. The first considered approach is based on two parallel semi-implicit solutions with different commutation orders. The second estimator, called the “double extrapolation” method, is a modification of the embedded method approach. The introduction of the double extrapolation LTE estimator allowed us to additionally increase the precision of the ESIMM solver. Using several known nonlinear systems, including stiff van der Pol oscillator, as the testbench, we explicitly show that ESIMM solvers can outperform both implicit and explicit linear multistep methods when implemented with an adaptive stepsize.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Adaptive Stepsize Control for Extrapolation Semi-Implicit Multistep ODE Solvers

Butusov

2021

Mathematics

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“…Let us consider a second-order initial value problem (IVP) of the form y (x) = f (x, y(x), y (x)), y(x 0 ) = y 0 , y (x 0 ) = y 0 , (1) on an interval [x 0 , b] ⊂ R, for which we assume that there exist a unique solution.…”

Section: Introductionmentioning

confidence: 99%

Efficient k-Step Linear Block Methods to Solve Second Order Initial Value Problems Directly

Ramos

Jator

Modebei³

2020

Mathematics

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There are dozens of block methods in literature intended for solving second order initial-value problems. This article aimed at the analysis of the efficiency of k-step block methods for directly solving general second-order initial-value problems. Each of these methods consists of a set of 2k multi-step formulas (although we will see that this number can be reduced to k+1 in case of a special equation) that provides approximate solutions at k grid points at once. The usual way to obtain these formulas is by using collocation and interpolation at different points, which are not all necessarily in the mesh (it may also be considered intra-step or off-step points). An important issue is that for each k, all of them are essentially the same method, although they can adopt different formulations. Nevertheless, the performance of those formulations is not the same. The analysis of the methods presented give some clues as how to select the most appropriate ones in terms of computational efficiency. The numerical experiments show that using the proposed formulations, the computing time can be reduced to less than half.

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“…These are problems of the form

y^{'^{'}} (x) = f (x, y (x)), x \in [x_{0}, X], y (x_{0}) = y_{0}, y^{'} (x_{0}) = y_{0}^{'} .

Many categories of numerical methods have been developed for the numerical solution of this problem, among them are multistep methods and Runge–Kutta–Nyström methods. Particularly, some modified Falkner‐type methods for this special second‐order initial‐value problem have been presented .…”

Section: Introductionmentioning

confidence: 99%

“…Many categories of numerical methods have been developed for the numerical solution of this problem, among them are multistep methods and Runge-Kutta-Nyström methods. Particularly, some modified Falkner-type methods for this special second-order initialvalue problem have been presented [1,2]. Special multistep methods for second-order ODEs have been considered in the literature, the well-known Numerov method is a classical example.…”

Section: Introductionmentioning

confidence: 99%

Modified two‐step hybrid methods for the numerical integration of oscillatory problems

Monovasilis¹,

Kalogiratou²,

Ramos

et al. 2017

Math Methods in App Sciences

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The construction of modified two‐step hybrid methods for the numerical solution of second‐order initial value problems with periodic or oscillatory behavior is considered. The coefficients of the new methods depend on the frequency of each problem so that the harmonic oscillator y′′=−w2y is integrated exactly. Numerical experiments indicate that the new methods are more efficient than existing methods with constant or variable coefficients. Copyright © 2017 John Wiley & Sons, Ltd.

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An efficient variable step-size rational Falkner-type method for solving the special second-order IVP

Cited by 17 publications

References 11 publications

Adaptive Stepsize Control for Extrapolation Semi-Implicit Multistep ODE Solvers

Adaptive Stepsize Control for Extrapolation Semi-Implicit Multistep ODE Solvers

Efficient k-Step Linear Block Methods to Solve Second Order Initial Value Problems Directly

Modified two‐step hybrid methods for the numerical integration of oscillatory problems

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