Semiconductor equations
Analysis of a one-dimensional model for the immersed boundary method
SIAM Journal on Numerical Analysis
Multi-phase computations in geometrical optics
Journal of Computational and Applied Mathematics - Special issue on TICAM symposium
Motion of curves in three spatial dimensions using a level set approach
Journal of Computational Physics
High-frequency wave propagation by the segment projection method
Journal of Computational Physics
Using K-branch entropy solutions for multivalued geometric optics computations
Journal of Computational Physics
Geometric optics in a phase-space-based level set and Eulerian framework
Journal of Computational Physics
A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations
Mathematics of Computation
A Slowness Matching Eulerian Method for Multivalued Solutions of Eikonal Equations
Journal of Scientific Computing
Regularization Techniques for Numerical Approximation of PDEs with Singularities
Journal of Scientific Computing
Journal of Computational Physics
Numerical approximations of singular source terms in differential equations
Journal of Computational Physics
Discretization of Dirac delta functions in level set methods
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
A Level Set Framework for Capturing Multi-Valued Solutions of Nonlinear First-Order Equations
Journal of Scientific Computing
Computing multi-valued velocity and electric fields for 1D Euler--Poisson equations
Applied Numerical Mathematics
Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation
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
Numerical Approximation of the Euler-Poisson-Boltzmann Model in the Quasineutral Limit
Journal of Scientific Computing
Hi-index | 31.45 |
We present a field-space-based level set method for computing multi-valued solutions to one-dimensional Euler-Poisson equations. The system of these equations has many applications, and in particular arises in semiclassical approximations of the Schrodinger-Poisson equation. The proposed approach 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 presented and compared with some exact solutions which demonstrate the effectiveness of the proposed method.