Integrating Proper Orthogonal Decomposition with the Crank–Nicolson Finite Element Method for Efficient Solutions of the Schrödinger Equation

This study proposes a dimensionality reduction method based on proper orthogonal decomposition (POD) for numerically solving the Schrödinger equation by optimizing the coefficient vectors of the Crank–Nicolson finite element (CNFE) solution. The fully discrete CNFE scheme is derived from the Schrödinger equation, with the stability and convergence of the CNFE solution rigorously established. Utilizing the POD technology, POD basis functions are constructed and a reduced-dimension model is formulated. The uniqueness and unconditional stability are proved, and the error estimate of the reduced-dimension solution is derived. With a 100×100 spatial grid, the reduced-dimension CNFE (RDCNFE) method employing POD technology reduces the degrees of freedom each time step from 1012 to 6. Numerical results show that for each additional second of simulation time, the CPU runtime of the standard CNFE method increases by several seconds, while the RDCNFE method increase remains below one second. This demonstrates that the reduced-dimension method significantly enhances computational efficiency for the Schrödinger equation while preserving numerical accuracy.