Gaussian Beam Methods for the Schrödinger Equation in the Semi-classical Regime: Lagrangian and Eulerian Formulations

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.