A Bloch band based level set method for computing the semiclassical limit of Schrِdinger equations

A novel Bloch band based level set method is proposed for computing the semiclassical limit of Schrödinger 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 eigenvalue problem and an associated Hamilton–Jacobi system for the phase in each Bloch band, with the Bloch eigenvalue be part of the Hamiltonian. We formulate a level set description to capture multi-valued 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 multi-valued 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 steps (ii) and (iii), respectively. Numerical examples with different number of bands are shown to demonstrate the capacity of the method.