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<p>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.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年8月5日

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  • Valmir Ganiu,
  • Pascal Loesing,
  • Matthias Jaeger,
  • Dirk Schulz

摘要

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.