Computing multivalued physical observables for the semiclassical limit of the Schrِdinger equation

We develop a level set method for the computation of multivalued physical observables (density, velocity, etc.) for the semiclassical limit of the Schrödinger equation. This method uses an Eulerian formulation and applies directly to arbitrary number of space dimensions. The main idea is to evolve the density near an n-dimensional manifold that is identified as the common zeros of n level set functions in phase space. These level set functions are generated from solving the Liouville equation with initial data chosen to embed the phase gradient. Simultaneously we track a new quantity f by solving again the Liouville equation near the obtained zero level set but with initial density as initial data. The multivalued density and higher moments are thus resolved by integrating f along the n-dimensional manifold in the phase directions. We show that this is equivalent to using the Wigner approach but decomposing the velocity from the density, each of which evolves by the same Liouville equation. The main advantages of this approach, in contrast to the standard kinetic equation approach using the Liouville equation with a Dirac measure initial data, include: (1) the Liouville equations are solved with L∞ initial data, and a singular integral involving the Dirac-δ function is evaluated only in the post-processing step, thus avoiding oscillations and excessive numerical smearing; (2) a local level set method can be utilized to significantly reduce the computation in the phase space. These advantages allow us to compute, for the first time, all physical observables for multidimensional problems in an Eulerian framework.