Symmetry (Jan 2024)

Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem

  • Chein-Shan Liu,
  • Chih-Wen Chang,
  • Chung-Lun Kuo

DOI
https://doi.org/10.3390/sym16010120
Journal volume & issue
Vol. 16, no. 1
p. 120

Abstract

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In this paper, some one-step iterative schemes with memory-accelerating methods are proposed to update three critical values f′(r), f″(r), and f‴(r) of a nonlinear equation f(x)=0 with r being its simple root. We can achieve high values of the efficiency index (E.I.) over the bound 22/3=1.587 with three function evaluations and over the bound 21/2=1.414 with two function evaluations. The third-degree Newton interpolatory polynomial is derived to update these critical values per iteration. We introduce relaxation factors into the Dzˇunic´ method and its variant, which are updated to render fourth-order convergence by the memory-accelerating technique. We developed six types optimal one-step iterative schemes with the memory-accelerating method, rendering a fourth-order convergence or even more, whose original ones are a second-order convergence without memory and without using specific optimal values of the parameters. We evaluated the performance of these one-step iterative schemes by the computed order of convergence (COC) and the E.I. with numerical tests. A Lie symmetry method to solve a second-order nonlinear boundary-value problem with high efficiency and high accuracy was developed.

Keywords