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An alternating direction implicit method for 2D nonlinear Schrödinger equation with accelerated evaluation of Caputo derivative

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  • Dwivedi, Himanshu Kumar
  • Rajeev,
  • Zeng, Shengda

Abstract

We propose an efficient time-space discretization for nonlinear fractional Schrödinger equations involving Caputo tempered derivatives. A new tempered Alikhanov scheme with parameter λ is introduced, together with a fast sum-of-exponentials (SOE) implementation, reducing complexity to O(MKtlogKt) and memory to O(MlogKt). Spatial derivatives are approximated using a compact scheme, and an alternating direction implicit formulation is derived with perturbation terms for stability. A graded time mesh resolves the initial singularity, while adaptive time-stepping ensures long-time efficiency. Stability and maximum-norm error bounds are established via a discrete Grönwall inequality. Numerical tests confirm the theoretical convergence and demonstrate substantial savings in CPU time and storage over classical methods. This work presents a novel nonuniform tempered Alikhanov time-stepping framework for nonlinear tempered fractional Schrödinger equation(NL-TFSEs), combining robustness, high accuracy, and computational scalability.

Suggested Citation

  • Dwivedi, Himanshu Kumar & Rajeev, & Zeng, Shengda, 2026. "An alternating direction implicit method for 2D nonlinear Schrödinger equation with accelerated evaluation of Caputo derivative," Applied Mathematics and Computation, Elsevier, vol. 518(C).
    • Handle: RePEc:eee:apmaco:v:518:y:2026:i:c:s0096300325006307
    DOI: 10.1016/j.amc.2025.129905
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    1. Dwivedi, Himanshu Kumar & Rajeev,, 2025. "A novel fast second order approach with high-order compact difference scheme and its analysis for the tempered fractional Burgers equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 227(C), pages 168-188.
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