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In this thesis, we propose one approach to compute the multi-valued density via the existing level set method in computing multi-valued velocity of WKB system in high frequency wave dynamics and a superposition property is proved and numerically validated, too. Several novel level set methods are developed and analyzed for computing multi-valued solutions to the Euler-Poisson equations and the Schrödinger equation with periodic structures. For the WKB system arising in high frequency wave dynamics, we propose a approach for computing the multi-valued density, which is useful for reconstruction of the original wave field. We also show that physical observables evaluated in Jin et al. (2005a,c) are simply the superposition of their multi-valued correspondents. Our method applies to the wave fields in both the Schrödinger equation and the optical wave equation. For these two applications a series of numerical tests is performed to compute multi-valued quantities and validate the established superposition properties. For one-dimensional Euler-Poisson equations, we propose a novel level set method. The method involves an implicit Eulerian formulation in an augmented space—called field space, which incorporates both velocity and electric fields into the configuration. Both velocity and electric fields are captured through common zeros of two level set functions, which are governed by a field transport equation. Simultaneously we obtain a weighted density f by solving again the field transport equation but with initial density as starting data. The averaged density is then resolved by the integration of the obtained f against the Dirac delta-function of two level set functions in the field space. Moreover, we prove that such obtained averaged density is simply a linear superposition of all multi-valued densities; and the averaged field quantities are weighted superposition of corresponding multi-valued ones. Computational results are