Symmetric collocation ERKN methods for general second-order oscillators

You, Xiong; Zhang, Ruqiang; Huang, Ting; Fang, Yonglei

doi:10.1007/s10092-019-0344-1

Cited by 4 publications

(5 citation statements)

References 18 publications

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“…with local truncation error LTE = − 17 90128941056000 h 11 𝑦 (11) (x 0 )+O ( h 12 ) has been used to estimate the local error at each step and a similar strategy for mesh selection as given above with p = 10 has also been utilized with the Runge-Kutta method.…”

Section: Error Estimation and Mesh Selectionmentioning

confidence: 99%

“…Many existing numerical methods for solving the class of problem in (1) have been developed; see, for example, previous studies. [2][3][4][5][6][7][8][9][10][11][12] Those strategies include Runge-Kutta type, linear multistep, Numerov-type, P-stable Obrechkoff, or collocation methods. One standard approach is to transform problem (1) into an equivalent system of first-order ODEs.…”

Section: Introductionmentioning

confidence: 99%

“…Many existing numerical methods for solving the class of problem in () have been developed; see, for example, previous studies 2–12 . Those strategies include Runge‐Kutta type, linear multistep, Numerov‐type, P‐stable Obrechkoff, or collocation methods.…”

Section: Introductionmentioning

confidence: 99%

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An adaptive optimized Nyström method for second‐order IVPs

Rufai

Mazzia

Ramos

2022

Math Methods in App Sciences

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This research work deals with the development, analysis, and implementation of an adaptive optimized one-step Nyström method for solving second-order initial value problems of ODEs and time-dependent partial differential equations. The new method is developed through a collocation technique with a new approach for selecting the collocation points. An embedding-like procedure is used to estimate the error of the proposed optimized method. The current approach has been used to compute efficiently approximate solutions to general second-order IVPs. The numerical experiments demonstrate that the introduced error estimation and step-size control strategy presented in this manuscript have produced a good performance compared to some of the other existing numerical methods. KEYWORDS collocation method, error estimation and control, ordinary and time-dependent partial differential equations, optimized Nyström method, variable stepsize formulation MSC CLASSIFICATION 65L05, 65L20

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Section: Error Estimation and Mesh Selectionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An adaptive optimized Nyström method for second‐order IVPs

Rufai

Mazzia

Ramos

2022

Math Methods in App Sciences

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“…Since the analytic solution of this problem does not exists, we used a reference numerical solution which was obtained by Anderson and Geer [2] and Verhulst [45]. The Maximum global error, − log 10 ( y (x) − y n ∞ ), of the FFBNM compared with the Gauss methods studied in [49] are displayed in Table 6, while the efficiency curves are shown in Fig. 8, respectively.…”

Section: Examplementioning

confidence: 99%

“…Many prominent authors including Simos [39], Coleman and Duxbury [6], Achar [3], Franco [14][15][16][17], Wang et al [46], Van Daele et al [43], Wang et al [47,48], Fang et al [8][9][10][11]49], Tsitouras [41], Li et al [30], Jator et al [21][22][23][24][25][26][27], Ramos et al [32][33][34][35][36][37][38]40], among others, have presented numerical methods for directly solving the equation in (1.1) without transforming it into an equivalent system of first-order ODEs. However, the above contributions did not consider that the approximate interpolating function was a linear combination of monomial, trigonometric terms and hyperbolic terms.…”

Section: Introductionmentioning

confidence: 99%

A Functionally-Fitted Block Numerov Method for Solving Second-Order Initial-Value Problems with Oscillatory Solutions

et al. 2021

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A functionally-fitted Numerov-type method is developed for the numerical solution of second-order initial-value problems with oscillatory solutions. The basis functions are considered among trigonometric and hyperbolic ones. The characteristics of the method are studied, particularly, it is shown that it has a third order of convergence for the general second-order ordinary differential equation, $$y''=f \left( x,y,y' \right) $$ y ′ ′ = f x , y , y ′ , it is a fourth order convergent method for the special second-order ordinary differential equation, $$y''=f \left( x,y\right) $$ y ′ ′ = f x , y . Comparison with other methods in the literature, even of higher order, shows the good performance of the proposed method.

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