A frequency-dependent \(p\)-adaptive technique for spectral methods. (English) Zbl 07516450
Summary: When using spectral methods, a consistent method for tuning the expansion order is often required, especially for time-dependent problems in which oscillations emerge in the solution. In this paper, we propose a frequency-dependent \(p\)-adaptive technique that adaptively adjusts the expansion order based on a frequency indicator. Using this \(p\)-adaptive technique, combined with recently proposed scaling and moving techniques, we are able to devise an adaptive spectral method in unbounded domains that can capture and handle diffusion, advection, and oscillations. As an application, we use this adaptive spectral method to numerically solve Schrödinger’s equation in an unbounded domain and successfully capture the solution’s oscillatory behavior at infinity.
MSC:
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 65Nxx | Numerical methods for partial differential equations, boundary value problems |
| 35Qxx | Partial differential equations of mathematical physics and other areas of application |
Keywords:
adaptive spectral method; Schrödinger equation; unbounded domains; Jacobi polynomial; Hermite function; Laguerre functionReferences:
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