Gaussian beam methods for the Schrodinger equation in the semi-classical regime: Lagrangian and Eulerian formulations

Abstract

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.

Gaussian beam methods for the Dirac equation in the semi-classical regime

“…Remark 3.2. As discussed in [10], to compute the Gaussian beam solutions for the Schrödinger Equation, the optimal mesh can be O(ε …”

“…The solution will be shown to have a good accuracy even around caustics, with a coarse mesh size of O( √ ε) and large time step of O( √ ε). A remarkable aspect of the Eulerian Gaussian beam method is that it still possesses the advantage of the previous method [10], which is an important benefit for the 3D simulation.…”

“…It is the goal of this paper to develop such a method by extending the previous method of [10] for the Schrödinger Equation to the Dirac Equation (1.1)-(1.2). With the help of the eigenvalue decomposition, the Gaussian beams evolve independently of each other.…”

“…Most of the methods were in the Lagrangian framework. More recently, Eulerian Gaussian beam methods have also received special attention for their advantage of uniform accuracy [10,11,12,13,16,17]. A major simplification of the Eulerian Gaussian beam method is that the Hessian matrices can be constructed by taking derivatives of the level set functions [10].…”

Convergence of a semiclassical wavepacket based time-splitting for the Schrödinger equation

“…The wavepackets have already been useful in the time integration of the semiclassical time-dependent Schrödinger equation in many dimensions via a special Strang-splitting [5]. They are related to higher-order Gaussian beams which are known to allow computational meshes of size O(ε), and have several appealing properties, see e.g., [8,9,14]. A family of wavepackets forms an orthonormal basis of L 2 , which gives them several advantages.…”