Hybrid Discontinuous Galerkin Approach for the Solution of Quantum Liouville-Type Equations

Efficiency Analysis of Discontinuous Galerkin Approaches for the Application Onto Quantum-Liouville Type Equations

“…In [11], the FV method was utilized in the ξ-direction as to better asses the influence of the numerical flux, introduced by the DG scheme in the χ-direction, on the statistical density. Since this was thoroughly analyzed in [10], the DG algorithm can now be introduced to discretize the ξ-domain according to Frensley's concept [14].…”

“…To find a discrete solution for each element, test functions l i,j (σ) must be specified. These functions are piece-wise defined polynomials of order N = N p − 1 and form a N -dimensional finite element [11]. Next, the test functions are integrated over each element D k .…”

“…For the discretization of the χ-coordinate in (10), the DG algorithm is utilized as depicted in [11] and shall be evaluated briefly in the following. The computational domain Ω χ within the interval [0, L χ ], is subdivided into a number N χ of finite elements.…”

“…In a next step, the term containing the χ-derivative is partially integrated two times to arrive at the strong formulation. Together with the relationship (2) and ( 3), (10) can be rewritten to [11]…”

“…Therefore, it is expedient to simulate nano devices on the basis of Quantum-Liouville-tye equations in center of mass coordinates with the DG algorithm. As proposed in [11], a hybrid DG method has already been proven to be a stable algorithm and deliver accurate results when simulating nano structures. However, the proposed method is of a hybrid nature by utilizing the FV technique to approximate the relative coordinate and the DG scheme to handle the center of mass coordinate.…”