Two-step Runge–Kutta methods for Volterra integro-differential equations

Abstract

In this paper, we investigate two-step Runge–Kutta methods to solve Volterra integro-differential equations. Two-step Runge–Kutta methods increase the order of convergence in comparing the classical Runge–Kutta method without extra computational cost. First, the local order conditions and convergence theorem are derived. Then, stability properties of two-step Runge–Kutta methods corresponding to the basic and convolution test equations are analysed. Furthermore, one-stage method with order four and two-stage method with order six are constructed and we plot the stability regions. Numerical examples are presented to confirm the theoretical analyses.

KEYWORDS:

• Volterra integro-differential equation
• two-step Runge–Kutta methods
• convergence
• stability

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65R20
• 65L06

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [No.12171177], National Natural Science Foundations of Henan Province [Nos. 222300420585, 232300420117], PhD Start-up Project of ZZULI [No. 2021BSJJ037].