Physical and numerical viscosity for quantum hydrodynamics

Communications in Mathematical Sciences

Volumn 5, Issue 2, 2007, Pages 447-471

  Jüngel, Ansgar ^a   Milišić, Josipa Pina ^b  

^a VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

^b JOHANNES GUTENBERG UNIVERSITY   (Germany)

Author keywords

Existence of stationary solutions; Numerical dispersion; Numerical viscosity; Quantum hydrodynamics; Resonant tunnelling diode; Semiconductors; Viscous quantum hydrodynamics

Indexed keywords

EID: 34547218452     PISSN: 15396746     EISSN: None     Source Type: Journal    
DOI: 10.4310/CMS.2007.v5.n2.a11     Document Type: Article

References (35)

1. A. Arnold and A. Jüngel, Multi-scale modeling of quantum semiconductor devices, A. Mielke (ed.), Analysis, Modeling and Simulation of Multiscale Problems, Springer, Berlin, 331-363, 2006.
2. A. Arnold, J. Lopez, P. Markowich and J. Soler, An analysis of quantum Fokker-Planck models: a Wigner function approach, Rev. Mat. Iberoam., 20, 771-814, 2004.
3. R. C. Bowen, G. Klimeck, R. Lake, W. Frensley and T. Moise, Quantitative simulation of resonant tunneling diode, J. Appl. Phys., 81, 3207-3213, 1997.
4. D. Bresch, B. Desjardins and C. Lin, On some compressible fluid models: Korteweg, lubrication and shallow water systems, Commun. Part. Diff. Eqs., 28, 1009-1037, 2003.
5. A. Caldeira and A. Leggett, Path integral approach to quantum Brownian motion, Phys. A, 121A, 587-616, 1983.
6. F. Castella, L. Erdös, F. Frommlet and P. Markowich, Fokker-Planck equations as scaling limits of reversible quantum systems, J. Stat. Phys., 100, 543-601, 2000.
7. Z. Chen, A finite element method for the quantum hydrodynamic model for semiconductor devices, Comput. Math. Appl., 31, 17-26, 1996.
8. Z. Chen, B. Cockburn, C. Gardner and J. Jerome, Quantum hydrodynamic simulation of hysteresis in the resonant tunneling diode, J. Comput. Phys., 117, 274-280, 1995.
9. P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112, 587-628, 2003.
10. J. Dennis and R. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, Philadelphia, 1996.
11. A. El Ayyadi and A. Jüngel, Semiconductor simulations using a coupled quantum drift-diffusion Schrödinger-Poisson model, SIAM J. Appl. Math., 66, 554-572, 2005.
12. D. Ferry and J.-R. Zhou, Form of the quantum potential for use in hydrodynamic equations for semiconductor device modeling, Phys. Rev. B, 48, 7944-7950, 1993.
13. I. Gamba and A. Jüngel, Positive solutions of singular equations of second and third order for quantum fluids, Archive Rat. Mech. Anal., 156, 183-203, 2001.
14. C. L. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math., 54, 409-427, 1994.
15. C. L. Gardner and C. Ringhofer, Smooth quantum potential for the hydrodynamic model, Phys. Rev. E, 53, 157-167, 1996.
16. C. L. Gardner, G. Klimeck and C. Ringhofer, Smooth quantum hydrodynamic model vs. NEMO simulation of resonant tunneling diodes, J. Comput. Electr., 3, 95-102, 2004.
17. C. L. Gardner, Private communication, 2007.
18. I. Gasser, P. A. Markowich, D. Schmidt and A. Unterreiter, Macroscopic theory of charged quantum fluids, in P. Marcati (ed.), Mathematical Problems in Semiconductor Physics, Pitman, Rome, Italy, 340, 42-75, 1995.
19. I. Gasser and P. A. Markowich, Quantum hydrodynamics, Wigner transforms and the classical limit, Asympt. Anal., 14, 97-116, 1997.
20. M. P. Gualdani and A. Jüngel, Analysis of the viscous quantum hydrodynamic equations for semiconductors, Europ. J. Appl. Math., 15, 577-595, 2004.
21. M. P. Gualdani, A. Jüngel and G. Toscani, Exponential decay in time of solutions of the viscous quantum hydrodynamic equations, Appl. Math. Lett., 16, 1273-1278, 2003.
22. M. T. Gyi and A. Jüngel, A quantum regularization of the one-dimensional hydrodynamic model for semiconductors, Adv. Diff. Eqs., 5, 773-800, 2000.
23. A. Jüngel, A steady-state quantum Euler-Poisson system for semiconductors, Commun. Math. Phys., 194, 463-479, 1998.
24. A. Jüngel and H.-L. Li, Quantum Euler-Poisson system: global existence and exponential decay, Quart. Appl. Math., 62, 569-600, 2004.
25. A. Jüngel, M. C. Mariani and D. Rial, Local existence of solutions to the transient quantum hydrodynamic equations, Math. Models Meth. Appl. Sci., 12, 485-495, 2002.
26. A. Jüngel, D. Matthes and P. Milišić, Derivation of new quantum hydrodynamic equations using entropy minimization, SIAM J. Appl. Math., 67, 46-68, 2006.
27. A. Jüngel and J.-P. Milišić, Macroscopic quantum models with and without collisions, Proceedings of the Sixth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics, Kyoto, Japan., Bulletin Inst. Math., Academia Sinica, to appear, 2007.
28. A. Jüngel and S. Tang, Numerical approximation of the viscous quantum hydrodynamic model for semiconductors, Appl. Numer. Math., 56, 899-915, 2006.
29. G. Klimeck, Quantum and semi-classical transport in NEMO 1-D, J. Comput. Electr., 2, 177-182, 2003.
30. N. Kluksdahl, A. Kriman, D. Ferry and C. Ringhofer, Self-consistent study of the resonant-tunneling diode, Phys. Rev. B, 39, 7720-7735, 1989.
31. H.-L. Li and P. Marcati, Existence and asymptotic behavior of multi-dimensional quantum hydrodynamic model for semiconductors, Commun. Math. Phys., 245, 215-247, 2004.
32. E. Madelung, Quantentheorie in hydrodynamischer Form, Z. Physik, 40, 322-326, 1927.
33. P. Pietra and C. Pohl, Weak limits of the quantum hydrodynamic model, Proceedings of the International Workshop on Quantum Kinetic Theory, Breckenridge, Colorado, USA, Special Issue of VLSI Design, 9, 427-434, 1999.
34. P. Shukla and L. Stenflo, Stimulated scattering instabilities of electronmagnetic waves in an ultracold quantum plasma, Phys. Plasmas, 13, 044505-044508, 2006.
35. E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40, 749-759, 1932.