Abstract
The time-dependent Schrödinger equation can be solved using the Houbolt difference scheme or the space-time polynomial particular solutions method, with the former performing well in dissipative problems and the latter being suitable for wave-like problems. To handle variations in time-step sizes while overcoming numerical oscillations in non-smooth regions, these two methods are coupled. The time-domain is handled using the Houbolt difference format, while the spatial domain is handled using the method of polynomial particular solutions. The numerical results show that the advantages of each method are preserved. Additionally, this hybrid approach can be replaced with other difference methods and meshless methods to meet various numerical solution needs.
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