The solution to high frequency waves (acoustic waves, elastic waves, quantum mechanics, electromagnetic waves, etc.) is computationally challenging due to the small wave length \epsilon.
A direct numerical simulation requires the mesh size (and time step) to be O(\epsilon). The Gaussian beam method is an efficient way to solve the high frequency wave equations asymptotically, outperforming geometric optics method in that the Gaussian beam method is accurate even at caustics.
We describe our Gaussian beam method using the example of the Schrodinger equation in the semiclassical regime, where the scaled Planck constant epsilon is extremely small. Our 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.
This is a joint work with Xu Yang and Hao Wu.