[Paper Review] Frozen Gaussian approximation for general linear strictly hyperbolic system: formulation and Eulerian methods

Jianfeng Lu, Xu Yang|arXiv (Cornell University)|Oct 10, 2010

Quantum chaos and dynamical systems12 references4 citations

TL;DR

This paper extends the frozen Gaussian approximation (FGA) to general linear strictly hyperbolic systems and develops Eulerian numerical methods to overcome the trajectory divergence problem inherent in Lagrangian FGA. By formulating FGA in an Eulerian framework using Liouville equations on phase space, the method enables stable, high-accuracy computation of high-frequency wave propagation, even in the presence of caustics and beam spreading, with applications to both wave equations and the Herman-Kluk propagator in quantum mechanics.

ABSTRACT

The frozen Gaussian approximation, proposed in [Lu and Yang, [15]], is an efficient computational tool for high frequency wave propagation. We continue in this paper the development of frozen Gaussian approximation. The frozen Gaussian approximation is extended to general linear strictly hyperbolic systems. Eulerian methods based on frozen Gaussian approximation are developed to overcome the divergence problem of Lagrangian methods. The proposed Eulerian methods can also be used for the Herman-Kluk propagator in quantum mechanics. Numerical examples verify the performance of the proposed methods.

Motivation & Objective

• Address the instability of Lagrangian frozen Gaussian approximation due to particle trajectory divergence in long-time high-frequency wave propagation.
• Generalize the frozen Gaussian approximation framework to general linear strictly hyperbolic systems with smooth coefficients.
• Develop Eulerian numerical methods based on phase-space Liouville equations to enable stable computation on fixed grids.
• Enable application of the method to the Herman-Kluk propagator in quantum mechanics for high-frequency Schrödinger equations.
• Provide a numerically robust alternative to conventional methods that require fine meshing proportional to wavelength.

Proposed method

• Extend FGA to M×M linear hyperbolic systems by defining a phase-space representation using Gaussian functions with frozen width.
• Formulate the evolution of the frozen Gaussian beam in the Eulerian framework via the Liouville equation on phase space.
• Solve the Liouville equation locally on a fixed phase-space grid to avoid particle trajectory divergence.
• Use semi-Lagrangian and finite-difference schemes for efficient time integration of the Liouville equation.
• Reconstruct physical-space solutions from phase-space integrals using quadrature rules with appropriate mesh sizes.
• Apply the method to both acoustic wave equations and Schrödinger equations via the Herman-Kluk propagator.

Experimental results

Research questions

• RQ1 How can the frozen Gaussian approximation be generalized to general linear strictly hyperbolic systems beyond scalar wave equations?
• RQ2 What are the limitations of Lagrangian FGA in long-time high-frequency wave propagation, and how can they be overcome?
• RQ3 Can Eulerian methods based on phase-space dynamics provide stable and accurate solutions for high-frequency wave propagation?
• RQ4 To what extent can the Eulerian FGA method be applied to quantum mechanical systems, such as the Herman-Kluk propagator?
• RQ5 How does the performance of Eulerian FGA compare to Lagrangian FGA and spectral methods in terms of accuracy and stability under beam spreading or caustics?

Key findings

• The Eulerian FGA method successfully resolves the divergence problem of Lagrangian FGA by using fixed-grid phase-space evolution.
• Numerical results for a 2D acoustic wave system show good agreement with spectral solutions, with relative errors in pressure and velocity below 5% at T=1.0 using ε=1/64.
• For the 1D Schrödinger equation with spreading wave packets, the Eulerian method maintains accuracy beyond T=10, while the Lagrangian method fails due to trajectory divergence.
• In the 2D Schrödinger equation with harmonic potential, the Eulerian Herman-Kluk propagator accurately captures both spreading and localizing dynamics at T=0.5 and T=1.
• The method achieves high accuracy with moderate mesh sizes (e.g., δx=1/64, δq=δp=1/32), avoiding the need for wavelength-resolved grids.
• The proposed Eulerian FGA framework is directly applicable to the Herman-Kluk propagator, extending its utility to quantum dynamics with high-frequency initial data.