Abstract
In this paper, we investigate highly stable multistep Runge–Kutta methods for Volterra integral equations. First, the order conditions for order p and stage order \(q=p\) are presented, and a convergence theorem is given. The numerical stability conditions for the basic and convolution test equations are derived. Then, the methods with one or two stages are studied in detail. Some A-stable and \(V_0\)-stable m-stage methods with order \(p>m\) are obtained. For one-stage methods, we also construct \(A_0\)-stable and 
-stable methods of orders 3 and 4. Finally, numerical experiments are given to confirm the theoretical results.
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The authors are very grateful to the anonymous referees and the editors for their valuable suggestions and comments.
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This work was supported by National Natural Science Foundation of China (Nos. 11771163, 12011530058 and 12071403).
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Communicated by Hui Liang.
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Wen, J., Xiao, A. & Huang, C. Highly stable multistep Runge–Kutta methods for Volterra integral equations. Comp. Appl. Math. 39, 308 (2020). https://doi.org/10.1007/s40314-020-01351-z
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DOI: https://doi.org/10.1007/s40314-020-01351-z