Abstract. The solution to the Schrödinger equation is highly oscillatory when the rescaled Planck constant ε is small in the semiclassical regime. A direct numerical simulation requires the mesh size to be O(ε). The Gaussian beam method is an efficient way to solve the high frequency wave equations asymptotically, outperforming the geometric optics method in that the Gaussian beam method is accurate even at caustics.In this paper, we solve the Schrödinger equation using both the Lagrangian and Eulerian formulations of the Gaussian beam methods. A new Eulerian Gaussian beam method is developed using the level set method based only on solving the (complex-valued) homogeneous Liouville equations. A major contribution here is that we are able to construct the Hessian matrices of the beams by using the level set function's first derivatives. This greatly reduces the computational cost in computing the Hessian of the phase function in the Eulerian framework, yielding an Eulerian Gaussian beam method with computational complexity comparable to that of the geometric optics but with a much better accuracy around caustics.We verify through several numerical experiments that our Gaussian beam solutions are good approximations to Schrödinger solutions even at caustics. We also numerically study the optimal relation between the number of beams and the rescaled Planck constant ε in the Gaussian beam summation.

We propose the frozen Gaussian approximation for computation of high frequency wave propagation. This method approximates the solution to the wave equation by an integral representation. It provides a highly efficient computational tool based on the asymptotic analysis on phase plane. Compared to geometric optics, it provides a valid solution around caustics.Compared to the Gaussian beam method, it overcomes the drawback of beam spreading. We give several numerical examples to verify that the frozen Gaussian approximation performs well in the presence of caustics and when the Gaussian beam spreads. Moreover, it is observed numerically that the frozen Gaussian approximation exhibits better accuracy than the Gaussian beam method. Both authors are grateful to Weinan E for his inspiration and helpful suggestions and discussions that improve the presentation. J.L. would like to thank Lexing Ying for stimulating discussions and for providing preprints [22,23] before publication. Part of the work was done during J.L.'s visits to Institute of Computational Mathematics and Scientific/Engineering Computing of Chinese Academy of Sciences and Peking University, and X.Y.'s visits to Tsinghua University and Peking University. We appreciate their hospitality. We thank anonymous referees for their valuable suggestions and remarks.

The frozen Gaussian approximation provides a highly efficient computational method for high‐frequency wave propagation. The derivation of the method is based on asymptotic analysis. In this paper, for general linear strictly hyperbolic systems, we establish the rigorous convergence result for frozen Gaussian approximation. As a byproduct, higher‐order frozen Gaussian approximation is developed. © 2011 Wiley Periodicals, Inc.

A pathway-based mean-field theory (PBMFT) was recently proposed for E. coli chemotaxis in [G. Si, T. Wu, Q. Quyang and Y. Tu, Phys. Rev. Lett., 109 (2012), 048101]. In this paper, we derived a new moment system of PBMFT by using the moment closure technique in kinetic theory under the assumption that the methylation level is locally concentrated. The new system is hyperbolic with linear convection terms. Under certain assumptions, the new system can recover the original model. Especially the assumption on the methylation difference made there can be understood explicitly in this new moment system. We obtain the Keller-Segel limit by taking into account the different physical time scales of tumbling, adaptation and the experimental observations. We also present numerical evidence to show the quantitative agreement of the moment system with the individual based E. coli chemotaxis simulator.

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.