Development of Discontinuous Galerkin Methods Applied to the Liouville–von Neumann Equation
The study of transport behavior within components of nanoelectronics and nanophotonics places high demands on methodological accuracy to describe realistic scenarios based on coherent or incoherent effects. Building on the Liouville–von Neumann equation in center-of-mass coordinates, a finite-volume technique enables the numerical treatment of stationary and transient transport processes, including complex potentials for open structures. By choosing a suitable basis in the approximation, a phase-space representation can be used to capture scattering mechanisms. This permits a more flexible implementation of transient algorithms compared with nonequilibrium Green’s function approaches and avoids the drawbacks of common methods for solving the Wigner transport equation that rely on discrete Fourier transforms. The discontinuous Galerkin method as a numerical scheme has been advanced in many research fields in recent years because of its high order of approximation and numerical efficiency. This methodology has not yet been applied to the Liouville–von Neumann equation. Compared with existing methods, it would achieve further efficiency gains and a markedly improved description of many interaction effects, for which a pronounced reduction of numerical dispersion to avoid phase errors is essential. The aim is therefore to develop a discontinuous Galerkin method for solving the Liouville–von Neumann equation such that it can also be extended to other important applications, for example spintronics or phonon transport.
DFG Individual Research Grant
- Title: Development of Discontinuous Galerkin Methods Applied to the Liouville–von Neumann Equation
- Project number: 446321675
- Project Duration: funded since 2021
- Funding: Individual Research Grant
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