Stabilizing Quantum-Liouville equations with complex absorbing potentials in discontinuous Galerkin frameworks

Abstract Efficiently solving quantum transport equations requires numerical methods that can handle open boundary conditions while maintaining accuracy and stability. This work introduces a novel approach combining the Discontinuous Galerkin (DG) method with Complex Absorbing Potentials (CAP) to approximate the Liouville-von Neumann equation. The DG method is particularly suited for high-performance computing, leveraging its block-diagonal matrix structure for parallelization and computational efficiency. By integrating CAPs, we mitigate non-physical boundary reflections, thereby improving stability and accuracy in both stationary and transient regimes. We conduct a rigorous analysis to evaluate how the integration of a CAP influences the stability and coercivity of the DG scheme, validating its effectiveness through numerical experiments on a resonant tunneling diode. Results demonstrate that CAP significantly reduces oscillatory artifacts and stabilizes the DG scheme by ensuring the eigenvalue spectrum lies in the left half-plane. This approach provides a robust framework for quantum transport simulations in nanoscale devices.