An efficient ADI compact scheme with Strang splitting for the two-dimensional cubic-quintic-septic nonlinear Schrödinger equation

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Abstract

We studied the two-dimensional cubic-quintic-septic nonlinear Schrödinger equation. We first established its continuous mass, energy, and momentum conservation laws, and then proposed an alternating direction implicit (ADI) compact finite difference scheme combined with Strang splitting. The splitting technique isolates the nonlinear term for an exact analytical solution via phase rotation, while a 3-point Padé approximation resolves the linear term to fourth-order spatial accuracy without requiring boundary ghost points. We rigorously proved the scheme is unconditionally stable with a convergence rate of O(τ² + h⁴_x + h⁴_y). Numerical experiments validated the theoretical analysis and demonstrated the method’s high-order accuracy and efficiency.

Keywords

nonlinear Schrödinger equation / finite difference method / ADI method / Strang splitting

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