A Spectral Collocation Technique for the Solution of the Wigner–Poisson Problem
Copyright policy )A Spectral Collocation Technique for the Solution of the Wigner–Poisson Problem
A numerical method for solving the coupled Wigner-Poisson system of nonlinear pseudodifferential equations is proposed and its numerical properties are analyzed. The paper extends a known mixed difference- spectral method in the case of a known potential function in a straightforward way to computing the potential by solving the Poisson equation. However, the analysis of the method is restricted to the one- dimensional case with pure Dirichlet boundary conditions. The numerical method consists in a discretization of the Poisson equation by standard centered differences on a non-uniform grid, and a spectral method for the quantum Liouville equation based on a collocation approach on equidistant nodes. The resulting system of hyperbolic equations is discretized implicitly in time and using upwinding in space direction. The consistency of the method is analyzed in detail by deriving bounds for the local and global discretization errors, and the stability of the linearization is shown by using the boundedness of the discrete solution.
collocation, consistency, stability, Poisson equation, system of hyperbolic equations, quantum mechanical transport phenomena, quantum Liouville equation, Applications to the sciences, transport phenomena, Wigner-Poisson system, Finite difference methods for initial value and initial-boundary value problems involving PDEs, Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics, upwinding, Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs, spectral collocation method, nonlinear pseudodifferential equations, difference-spectral method, Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs, PDEs in connection with quantum mechanics
collocation, consistency, stability, Poisson equation, system of hyperbolic equations, quantum mechanical transport phenomena, quantum Liouville equation, Applications to the sciences, transport phenomena, Wigner-Poisson system, Finite difference methods for initial value and initial-boundary value problems involving PDEs, Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics, upwinding, Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs, spectral collocation method, nonlinear pseudodifferential equations, difference-spectral method, Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs, PDEs in connection with quantum mechanics
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