Frozen Gaussian Grid-point Correction For Semi-classical Schrödinger Equation
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911174951436288 |
|---|---|
| author | Chai, Lihui Deng, Zili |
| author_facet | Chai, Lihui Deng, Zili |
| contents | We propose an efficient reconstruction algorithm named the frozen Gaussian grid-point correction (FGGC) for computing the Schrödinger equation in the semi-classical regime using the frozen Gaussian approximation (FGA). The FGA has demonstrated its superior efficiency in dealing with semi-classical problems and high-frequency wave propagations. However, reconstructing the wave function from a large number of Gaussian wave-packets is typically computationally intensive. This difficulty arises because these wave-packets propagate along the FGA trajectories to non-grid positions, making the application of the fast Fourier transform infeasible. In this work, we introduce the concept of ``on-grid correction'' and derive the formulas for the least squares approximation of Gaussian wave-packets, and also provide a detailed process of the FGGC algorithm. Furthermore, we rigorously prove that the error introduced by the least squares approximation on each Gaussian wave-packet is independent of the semi-classical parameter $\varepsilon$. Numerical experiments show that the FGGC algorithm can significantly improve reconstruction efficiency while introducing only negligible error, making it a powerful tool for solving the semi-classical Schrödinger equation, especially in applications requiring both accuracy and efficiency. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21785 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Frozen Gaussian Grid-point Correction For Semi-classical Schrödinger Equation Chai, Lihui Deng, Zili Numerical Analysis We propose an efficient reconstruction algorithm named the frozen Gaussian grid-point correction (FGGC) for computing the Schrödinger equation in the semi-classical regime using the frozen Gaussian approximation (FGA). The FGA has demonstrated its superior efficiency in dealing with semi-classical problems and high-frequency wave propagations. However, reconstructing the wave function from a large number of Gaussian wave-packets is typically computationally intensive. This difficulty arises because these wave-packets propagate along the FGA trajectories to non-grid positions, making the application of the fast Fourier transform infeasible. In this work, we introduce the concept of ``on-grid correction'' and derive the formulas for the least squares approximation of Gaussian wave-packets, and also provide a detailed process of the FGGC algorithm. Furthermore, we rigorously prove that the error introduced by the least squares approximation on each Gaussian wave-packet is independent of the semi-classical parameter $\varepsilon$. Numerical experiments show that the FGGC algorithm can significantly improve reconstruction efficiency while introducing only negligible error, making it a powerful tool for solving the semi-classical Schrödinger equation, especially in applications requiring both accuracy and efficiency. |
| title | Frozen Gaussian Grid-point Correction For Semi-classical Schrödinger Equation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2504.21785 |