Abstract: A novel Bloch band based level set method is proposed for computing the semiclassical limit of Schrodinger equations in periodic media. For the underlying equation subject to a highly oscillatory initial data, a hybrid of the WKB approximation and homogenization leads to the Bloch eigen value problem and an associated HamiltonJacobi system for the phase in each Bloch band, with the Bloch eigen value be part of the Hamiltonian. We formulate a level set description to capture multivalued solutions to the band WKB system, and then evaluate total homogenized density over a sample set of bands. A superposition of band densities is established over all bands and solution branches when away from caustic points. The numerical approach splits the solution process into several parts: i) initialize the level set function from the band decomposition of the initial data; ii) solve the Bloch eigenvalue problem to compute Bloch waves; iii) evolve the band level set equation to compute multivalued velocity and density on each Bloch band; iv) evaluate the total position density over a sample set of bands using Bloch waves and band densities obtained in step ii) and iii), respectively. Numerical examples with different number of bands are provided to demonstrate the good quality of the method.
The results have been done in collaboration with Hailiang Liu.
