The solution to the Schrödinger equation is highly oscillatory when the rescaled
Planck constant $\varepsilon$ is small in the semiclassical regime. A direct numerical simulation requires the
mesh size to be $\emph{O}(\varepsilon)$. 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 $\varepsilon$ in the Gaussian beam
summation.