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<p>The aim of this paper is to design the explicit radial basis function (RBF) Runge–Kutta methods for the initial value problem. We construct the two-, three- and four-stage RBF Runge–Kutta methods based on the Gaussian RBF Euler method with the shape parameter. The analysis of the local truncation error shows that the <i>s</i>-stage RBF Runge–Kutta method could formally achieve order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(s+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, whereas the original <i>s</i>-stage Runge–Kutta method attains only order <i>s</i>. The proof for the convergence of those RBF Runge–Kutta methods follows. We then plot the stability region of each RBF Runge–Kutta method proposed and compare with the one of the correspondent Runge–Kutta method. Numerical experiments are provided to exhibit the improved behavior of the RBF Runge–Kutta methods over the standard ones.</p>

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Explicit radial basis function Runge–Kutta methods

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年6月23日

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  • Jiaxi Gu,
  • Xinjuan Chen,
  • Jae-Hun Jung

摘要

The aim of this paper is to design the explicit radial basis function (RBF) Runge–Kutta methods for the initial value problem. We construct the two-, three- and four-stage RBF Runge–Kutta methods based on the Gaussian RBF Euler method with the shape parameter. The analysis of the local truncation error shows that the s-stage RBF Runge–Kutta method could formally achieve order \(s+1\) s + 1 , whereas the original s-stage Runge–Kutta method attains only order s. The proof for the convergence of those RBF Runge–Kutta methods follows. We then plot the stability region of each RBF Runge–Kutta method proposed and compare with the one of the correspondent Runge–Kutta method. Numerical experiments are provided to exhibit the improved behavior of the RBF Runge–Kutta methods over the standard ones.