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Explicit radial basis function Runge-Kutta methods

Jiaxi Gu, Xinjuan Chen, Jae-Hun Jung

TL;DR

The paper develops explicit radial basis function Runge–Kutta (RBF–RK) methods by embedding Gaussian RBF interpolation into explicit RK stages with shape parameters $\epsilon$, achieving formal order $s+1$ for an $s$-stage method (up to four stages) through LTE cancellation. It provides convergence proofs under Lipschitz continuity and bounded $\epsilon$, derives stability functions involving $e^{\alpha z^2}$, and presents numerical experiments showing improved accuracy over standard RK schemes, including stiff and nonseparable problems. The work demonstrates that a small additional computational effort to tune $\epsilon$ yields higher-order accuracy and favorable stability properties, offering a practical approach to enhanced explicit time integration. Overall, the study highlights the potential of RBF-based temporal discretization to surpass classical RK performance while maintaining explicitness and tractable stability characteristics.

Abstract

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, where the analysis of the local truncation error shows that the s-stage RBF Runge-Kutta method could formally achieve order s+1. 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.

Explicit radial basis function Runge-Kutta methods

TL;DR

The paper develops explicit radial basis function Runge–Kutta (RBF–RK) methods by embedding Gaussian RBF interpolation into explicit RK stages with shape parameters , achieving formal order for an -stage method (up to four stages) through LTE cancellation. It provides convergence proofs under Lipschitz continuity and bounded , derives stability functions involving , and presents numerical experiments showing improved accuracy over standard RK schemes, including stiff and nonseparable problems. The work demonstrates that a small additional computational effort to tune yields higher-order accuracy and favorable stability properties, offering a practical approach to enhanced explicit time integration. Overall, the study highlights the potential of RBF-based temporal discretization to surpass classical RK performance while maintaining explicitness and tractable stability characteristics.

Abstract

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, where the analysis of the local truncation error shows that the s-stage RBF Runge-Kutta method could formally achieve order s+1. 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.
Paper Structure (16 sections, 3 theorems, 74 equations, 3 figures, 9 tables)

This paper contains 16 sections, 3 theorems, 74 equations, 3 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Explicit Runge-Kutta methods
  3. Two-stage second-order methods
  4. Three-stage third-order methods
  5. Four-stage fourth-order methods
  6. Explicit RBF Runge-Kutta methods
  7. Two-stage third-order methods
  8. Three-stage fourth-order methods
  9. Four-stage fifth-order methods
  10. Convergence of RBF Runge-Kutta methods
  11. Two-stage methods
  12. Three-stage methods
  13. Four-stage RBF Runge-Kutta methods
  14. Stability regions
  15. Numerical experiments
  16. ...and 1 more sections

Key Result

Theorem 4.1

Suppose that $\epsilon^2_n$ is bounded for all $n = 0, 1, \cdots, N-1$. Then the two-stage RBF Runge-Kutta method eq:rbf_rk2, satisfying the conditions eq:rk_consistent, eq:rk_cond and eq:rk2_cond, is convergent.

Figures (3)

  • Figure 1: Stability regions of RK2 (black) and RBF-RK2 (red).
  • Figure 2: Stability regions of RK3 (black) and RBF-RK3 (red, green, blue, cyan, magenta and purple).
  • Figure 3: Stability regions of RK4 (black) and RBF-RK4 (red, green, blue and cyan).

Theorems & Definitions (9)

  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • proof
  • Example 6.1
  • Example 6.2
  • Example 6.3