Frozen Gaussian Grid-point Correction for Semi-classical Schrödinger Equation

We propose an efficient reconstruction algorithm named the frozen Gaussian grid-point correction (FGGC) for computing solutions to the semi-classical Schrödinger equation based on the frozen Gaussian approximation (FGA). The FGA has demonstrated its superior efficiency in dealing with semi-classical problems and high-frequency wave propagation. However, reconstructing the wave function from a large number of Gaussian wave-packets is computationally intensive, as these wave-packets propagate along the FGA trajectories to off-grid positions, making the application of the fast Fourier transform infeasible. In this work, we introduce the concept of “on-grid correction”, derive formulas for the least squares approximation of Gaussian wave-packets, and provide a detailed description of the FGGC algorithm. Furthermore, we rigorously prove that the error introduced by the least squares approximation for each Gaussian wave-packet is independent of the semi-classical parameter \(\varepsilon \) $\epsilon$ . Numerical experiments show that the FGGC algorithm significantly improves 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.