Adaptive time-stepping scheme for the nonlinear Schrödinger equation with wave operator in a 2D unbounded domain: Convergence and conservation

Huiling Jiang and Dongdong Hu

Mathematics and Computers in Simulation (MATCOM), 2026, vol. 248, issue C, 497-513

Abstract: This paper is concerned with a novel linearly implicit and energy-preserving time-stepping scheme for the nonlinear Schrödinger equation with wave operator in two-dimensional unbounded domain. To this ends, we first reformulate the original equation in an equivalent modified system, and propose a second-order variable-step time-stepping scheme for the modified system by combining the Crank–Nicolson approach and linear interpolation. The proposed scheme is proven to be energy-preserving, uniquely solvable and convergent. Simultaneously, we choose the mapped Chebyshev spectral Galerkin method for the Laplacian to handle unbounded domain, and provide a fast implementation for the fully discrete scheme in detail. Extensive numerical comparisons between the uniform and adaptive time-stepping strategies show that the theoretical results are correct and the proposed scheme is highly efficient in long-time computation.

Keywords: Structure-preserving algorithm; Adaptive time-stepping strategy; Mapped Chebyshev spectral Galerkin method; Unbounded domain; Convergence; Unique solvability (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:248:y:2026:i:c:p:497-513

DOI: 10.1016/j.matcom.2026.04.031

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