Solving first-order initial-value problems by using an explicit non-standard  A -stable one-step method in variable step-size formulation

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

doi:10.1016/j.amc.2015.06.119

Cited by 10 publications

(4 citation statements)

References 21 publications

(31 reference statements)

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“…and employing both techniques as embedded type techniques. Same sort of approaches have also been used in [37,38]. No additional computational cost will be needed since the values required by lower order technique are also those to be employed by the higher order technique.…”

Section: Adaptive Step-size Approachmentioning

confidence: 99%

A New Nonlinear Hybrid Technique with fixed and adaptive step-size approaches

Soomro¹

2022

Sigma J Eng Nat Sci - Sigma Müh Fen Bil Derg

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Linear and nonlinear numerical techniques are the most popular techniques for finding approximate solutions to initial value problems in numerous scientific fields. Due to the substantial importance of ordinary differential equations, an attempt has been made in the present research study to obtain a new nonlinear hybrid technique based upon contraharmonic and harmonic means having fourth-order accuracy. Theoretical analysis in terms of consistency, stability, asymptotic errors (local and global truncation errors), and convergence has also been carried out. The newly formulated technique is compared with some existing techniques having the same characteristics and observed to be much better because of errors, CPU time, and stability region. The adaptive step-size approach improves the performance of the proposed technique, and strategies to control the errors are developed. Some numerical experiments for scalar and vector initial value problems, including logistic growth, sinusoidal and industrial Robot Arm systems, are presented to show better performance of the proposed technique.

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Section: Adaptive Step-size Approachmentioning

confidence: 99%

A New Nonlinear Hybrid Technique with fixed and adaptive step-size approaches

Soomro¹

2022

Sigma J Eng Nat Sci - Sigma Müh Fen Bil Derg

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“…To provide an alternative to these codes, we propose a class of unconventional explicit unconditionally stable numerical codes to accurately and efficiently integrate a given problem. Many authors have proposed these type of methods [23][24][25][26][27]. In this regard, we first present the derivation of the family of non-standard methods to integrate systems of ODEs arising in the semi-discretization of initial-boundary values of a Burgers' equation and analyse its basic characteristics, i.e., order of accuracy and linear stability analysis.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Non-Standard Numerical Scheme Coupled with a Compact Finite Difference Method to Solve the One-Dimensional Burgers’ Equation

Kaur

Singh

2023

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This article proposes a family of non-standard methods coupled with compact finite differences to numerically integrate the non-linear Burgers’ equation. Firstly, a family of non-standard methods is derived to deal with a system of ordinary differential equations (ODEs) arising from the semi-discretization of initial-boundary value partial differential equations (PDEs). Further, a method of this family is considered as a special case and coupled with a fourth-order compact finite difference resulting in a combined numerical scheme to solve initial-boundary value PDEs. The combined scheme has first-order accuracy in time and fourth-order accuracy in space. Some basic characteristics of the scheme are analysed and a section concerning the numerical experiments is presented demonstrating the good performance of the combined numerical scheme.

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“…The emergence of nonlinear methods in solving many kinds of problems under the scalar and vector versions has intrigued many researchers to study more on this topic. Some past and recently published research work on nonlinear methods are 7‐15 . The analysis of ordinary differential equations is an essential tool for investigating the relationship between various dynamical quantities on which more literature can be found in Reference 16.…”

Section: Introductionmentioning

confidence: 99%

A new family of 𝒜− acceptable nonlinear methods with fixed and variable stepsize approach

Qureshi

Soomro

Hınçal³

2021

Comp and Math Methods

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Solving stiff, singular, and singularly perturbed initial value problems (IVPs) has always been challenging for researchers working in different fields of science and engineering. In this research work, an attempt is made to devise a family of nonlinear methods among which second‐ to fourth‐order methods are not only 𝒜− stable but 𝒜− acceptable as well under order stars' conditions. These features make them more suitable for solving stiff and singular systems in ordinary differential equations. Methods with remaining orders are either zero‐ or conditionally stable. The theoretical analysis contains local truncation error, consistency, and order of accuracy of the proposed nonlinear methods. Furthermore, both fixed and variable stepsize approaches are introduced wherein the latter improves the performance of the devised methods. The applicability of the methods for solving the system of IVPs is also described. When used to solve problems from physical and real‐life applications, including nonlinear logistic growth and stiff model for flame propagation, the proposed methods are found to have good results.

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Solving first-order initial-value problems by using an explicit non-standard A -stable one-step method in variable step-size formulation

Cited by 10 publications

References 21 publications

A New Nonlinear Hybrid Technique with fixed and adaptive step-size approaches

A New Nonlinear Hybrid Technique with fixed and adaptive step-size approaches

An Efficient Non-Standard Numerical Scheme Coupled with a Compact Finite Difference Method to Solve the One-Dimensional Burgers’ Equation

A new family of 𝒜− acceptable nonlinear methods with fixed and variable stepsize approach

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