The stability of predictor-corrector methods of Runge-Kutta type for uncertain differential equations

Qiaohong Liu; Zhi Li; Liping Xu; Qiaohong Liu; Zhi Li; Liping Xu

doi:10.3934/math.2026479

AIMS Mathematics

2026, Volume 11, Issue 4: 11617-11633. doi: 10.3934/math.2026479

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Research article

The stability of predictor-corrector methods of Runge-Kutta type for uncertain differential equations

School of Information and Mathematics, Yangtze University, Jingzhou 434023, China

Received: 21 January 2026 Revised: 25 March 2026 Accepted: 31 March 2026 Published: 27 April 2026
MSC : 34D20, 65L20

This paper studies the asymptotically mean-square stability of a Runge–Kutta type predictor–corrector numerical scheme for uncertain differential equations. The method consists of a fourth-order Runge–Kutta predictor coupled with a one-step implicit correction, which leads to a fully discrete scheme for the associated $ \alpha $-path equations. For a linear test equation driven by a Liu process, the corresponding growth factor of the numerical solution is derived. By employing vectorization techniques and Kronecker product representations, a recursive relation is established for the second-moment matrix of the numerical solution. It is shown that the asymptotically mean-square stability of the proposed scheme is equivalent to a necessary and sufficient condition on the modulus of the growth factor. Numerical examples are provided to illustrate the theoretical results and to demonstrate the influence of the step size on the stability.
- uncertain differential equations,
- Runge-Kutta predictor-corrector method,
- asymptotically mean-square stability,
- Liu process,
- growth factor
Citation: Qiaohong Liu, Zhi Li, Liping Xu. The stability of predictor-corrector methods of Runge-Kutta type for uncertain differential equations[J]. AIMS Mathematics, 2026, 11(4): 11617-11633. doi: 10.3934/math.2026479

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Abstract

This paper studies the asymptotically mean-square stability of a Runge–Kutta type predictor–corrector numerical scheme for uncertain differential equations. The method consists of a fourth-order Runge–Kutta predictor coupled with a one-step implicit correction, which leads to a fully discrete scheme for the associated $ \alpha $-path equations. For a linear test equation driven by a Liu process, the corresponding growth factor of the numerical solution is derived. By employing vectorization techniques and Kronecker product representations, a recursive relation is established for the second-moment matrix of the numerical solution. It is shown that the asymptotically mean-square stability of the proposed scheme is equivalent to a necessary and sufficient condition on the modulus of the growth factor. Numerical examples are provided to illustrate the theoretical results and to demonstrate the influence of the step size on the stability.

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