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

Author

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  • Jiang, Huiling
  • Hu, Dongdong

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.

Suggested Citation

  • Jiang, Huiling & Hu, Dongdong, 2026. "Adaptive time-stepping scheme for the nonlinear Schrödinger equation with wave operator in a 2D unbounded domain: Convergence and conservation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 248(C), pages 497-513.
    • Handle: RePEc:eee:matcom:v:248:y:2026:i:c:p:497-513
    DOI: 10.1016/j.matcom.2026.04.031
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