A Discontinuous Galerkin Approach to Wigner's Equation

Richard Sharp
University of Texas
email: rsharp@math.utexas.edu

Abstract: The Wigner equation connects an isolated quantum system to external effects, which provides an appropriate setting for quantum device simulation. There are relatively few numerical simulations of Wigner's equation. Discrete and operator splitting methods have been produced, but each faces various limitations. We propose a discontinuous-Galerkin (DG) approach based on a non-polynomial approximation space. Yuan and Shu introduced the use of such approximation spaces to improve DG performance by reducing the complexity of the representation of the solution through careful selection of the basis. We will choose a basis based on numerical and physical considerations. Numerically, the goal is to use a small number of basis functions on a coarse mesh. A physically ideal situation exists (i.e. the solution over a mesh with one cell corresponding to the domain of the problem), but is not implementable. The practical approach will be to select basis functions that are both easy to implement and that provide economic representations of the physical solution. Examples include trigonometric functions in a region where the solution is oscillatory, or the eigenstates of an appropriate operator restricted to the mesh.