Computing multi-valued velocity and electric fields for 1D Euler--Poisson equations
Applied Numerical Mathematics
Journal of Computational Physics
Superposition of Multi-Valued Solutions in High Frequency Wave Dynamics
Journal of Scientific Computing
A Bloch band based level set method for computing the semiclassical limit of Schrödinger equations
Journal of Computational Physics
Journal of Computational Physics
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We introduce a level set method for the computation of multi-valued solutions of a general class of nonlinear first-order equations in arbitrary space dimensions. The idea is to realize the solution as well as its gradient as the common zero level set of several level set functions in the jet space. A very generic level set equation for the underlying PDEs is thus derived. Specific forms of the level set equation for both first-order transport equations and first-order Hamilton-Jacobi equations are presented. Using a local level set approach, the multi-valued solutions can be realized numerically as the projection of single-valued solutions of a linear equation in the augmented phase space. The level set approach we use automatically handles these solutions as they appear