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This paper explores the properties of explicitExplicit nonstandard Runge–Kutta (ENRK) methods for solving dynamical systemsSystems. It focuses on their application to higher-order methods that use distinct step-size functionsFunction for the main and internal stages. Standard explicitExplicit Runge–Kutta (SRK) methods often suffer from sensitivitySensitivity to the discretizationDiscretization step size, \(\Delta t\) , which can lead to a loss of dynamical consistency, particularly in long-term simulationsSimulation where stability and dynamical consistency are crucial. To overcome these challenges, we apply nonstandard finite difference (NSFD) techniques to improve numericalNumerical consistency. The study investigates the impact of different step-size functionsFunction on the preservation of essential numericalNumerical properties, such as the method’s order, consistency, and convergenceConvergence. Specifically, we identify key properties of the step-size functionsFunction that ensure theAnalysisExplicitComputationalAnalysisNumericalAnalysisResearchResearchConvergenceAnalysis ENRK method’s order is comparable to that of the corresponding SRK methods. Based on the findings, under certain conditionsConditions on the step-size functionsFunction, the order of consistency and convergenceConvergence of ENRK methods matches that of their corresponding SRK methods.

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Convergence Analysis of the Explicit Nonstandard Runge-Kutta Methods

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  • 发表日期：

    2026年

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  • Istvan Faragó,
  • Rahele Mosleh

摘要

This paper explores the properties of explicitExplicit nonstandard Runge–Kutta (ENRK) methods for solving dynamical systemsSystems. It focuses on their application to higher-order methods that use distinct step-size functionsFunction for the main and internal stages. Standard explicitExplicit Runge–Kutta (SRK) methods often suffer from sensitivitySensitivity to the discretizationDiscretization step size, \(\Delta t\) , which can lead to a loss of dynamical consistency, particularly in long-term simulationsSimulation where stability and dynamical consistency are crucial. To overcome these challenges, we apply nonstandard finite difference (NSFD) techniques to improve numericalNumerical consistency. The study investigates the impact of different step-size functionsFunction on the preservation of essential numericalNumerical properties, such as the method’s order, consistency, and convergenceConvergence. Specifically, we identify key properties of the step-size functionsFunction that ensure theAnalysisExplicitComputationalAnalysisNumericalAnalysisResearchResearchConvergenceAnalysis ENRK method’s order is comparable to that of the corresponding SRK methods. Based on the findings, under certain conditionsConditions on the step-size functionsFunction, the order of consistency and convergenceConvergence of ENRK methods matches that of their corresponding SRK methods.