Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime
Journal of Computational Physics
A Bloch band based level set method for computing the semiclassical limit of Schrödinger equations
Journal of Computational Physics
An Eulerian approach for computing the finite time Lyapunov exponent
Journal of Computational Physics
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We propose a local level set method for constructing the geometrical optics term in the paraxial formulation for the high frequency asymptotics of two-dimensional (2-D) acoustic wave equations. The geometrical optics term consists of two multivalued functions: a travel-time function satisfying the eikonal equation locally and an amplitude function solving a transport equation locally. The multivalued travel-times are obtained by solving a level set equation and a travel-time equation with a forcing term. The multivalued amplitudes are computed by a new Eulerian formula based on the gradients of travel-times and takeoff angles. As a byproduct the method is also able to capture the caustic locations. The proposed Eulerian method has complexity of $O(N^2{\rm Log }N)$, rather than $O(N^4)$ as typically seen in the Lagrangian ray tracing method. Several examples including the well-known Marmousi synthetic model illustrate the accuracy and efficiency of the Eulerian method.