Haifeng Sang, Tianliang Hou, A new unconditionally energy-stable finite difference scheme for Riesz space-fractional Allen-Cahn equations, Vol. 2026 (2026), No. 6, pp. 1-13

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DOI: 10.23952/jnfa.2026.6

Received August 27, 2025; Accepted January 8, 2026; Published February 4, 2026

Abstract. In this paper, we present a new second-order finite difference scheme for Riesz space-fractional Allen-Cahn equations. We use a modified Crank-Nicolson finite difference scheme with stabilized terms of third-order numerical accuracy for temporal discretization. The discrete maximum bound principle, the maximum-norm error estimates, and the discrete energy stability of the proposed scheme are discussed. It is demonstrated that the proposed scheme is maximum bound principle preserving and unconditionally energy-stable for any nonnegative stabilization parameter $\beta$ which satisfies $0\leq \beta\leq 1/4$ . As far as we know, the proposed fully implicit second-order scheme has never been proved to preserve the maximum bound principle before except the Allen-Cahn equation with $\beta=1/12$ and $\beta=0$ . Finally, some numerical experiments are performed to verify the theoretical results.

How to Cite this Article:
H. Sang, T. Hou, A new unconditionally energy-stable finite difference scheme for Riesz space-fractional Allen-Cahn equations, J. Nonlinear Funct. Anal. 2026 (2026) 6.