| Abstract: |
| We develop a novel class of high-order, structure-preserving methods for Schr\{o}dinger-type equations based on a predictor--corrector framework that integrates operator splitting with Lagrange multiplier corrections. The predictor employs high-order compact splitting spectral methods, in which the Hamiltonian is decomposed into two parts that are exactly solvable in either phase or physical space,
while the corrector enforces the exact preservation of multiple invariants through Lagrange multipliers. We rigorously prove the existence and uniqueness of the Lagrange multipliers under reasonable conditions, and show that the overall scheme retains the same temporal order as the underlying predictor.
The resulting schemes constitute the first class of split-type schemes capable of preserving multiple original invariants, and they demonstrate markedly improved long-time accuracy and stability compared to classical splitting approaches. Numerical experiments validate the theoretical findings and illustrate the effectiveness of the proposed methods for rotating dipolar Bose--Einstein condensates with two invariants and rotating spin-1 systems with three invariants. More broadly, the proposed framework provides a simple and general strategy for systematically transforming existing time integrators into structure-preserving methods for Hamiltonian systems. |
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