Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Introduction and Chronological Perspective Kapitel lesen Erstes Kapitel lesen

2013 | OriginalPaper | Buchkapitel

15. Balances in Two Dimensions: Kinetic Semiconductor Equations Again

verfasst von : Laurent Gosse

Erschienen in: Computing Qualitatively Correct Approximations of Balance Laws

Verlag: Springer Milan

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

The equations studied in Chapters 11 and 12, Vlasov-BGK and Vlasov-Fokker-Planck, are genuinely bi-dimensional problems. Thanks to their special structure, one can succeed in solving them by means of essentially one-dimensional algorithms because the formalism of elementary solutions can be extended up to some numerically tolerable approximations. Besides, it is somewhat tacitly assumed that one of the 2 directions of propagation is dominant. This leaves open the possibility of attacking these problems by means of truly bi-dimensional numerical schemes, treating the Vlasov acceleration term through a divided difference in the v direction, itself possibly containing a modified state which renders locally a source term’s effect. Kinetic problems are well suited for an investigation of bi-dimensional well-balanced discretizations also because the passage from one- to two-dimensional upwind schemes is generally associated to a change of paradigm: one switches from a nonlinear flux term like ∂ x f(u) to a linear advection equation ∂ t u + a∂ x u + b∂ y u = 0. Kinetic equations, like Vlasov equation (6.2), can be seen as being in midstream.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel Linearized BGK Model of Heat Transfer
Nächstes Kapitel Conclusion: Outlook and Shortcomings
Fußnoten
1
This weak sensibility to the grid parameters is reminiscent of the error estimate (3.22).
 
Literatur
1.
Aamodt R.E., Case K.M.: Useful identities for half-space problems in linear transport theory. Ann. Physics 21, 284–301 (1963)CrossRefMATHMathSciNet
2.
3.
Abgrall R., Mezine M.: Construction of second-order accurate monotone and stable residual distribution schemes for steady problems. J. Comput. Phys. 195, 474–507 (2004)CrossRefMATHMathSciNet
4.
Audusse E., Bristeau M.-O.: A well-balanced positivity preserving “second-order” scheme for shallow water flows on unstructured meshes. J. Comput. Phys. 206, 311–333 (2005)CrossRefMATHMathSciNet
5.
Balsara D.S., A two-dimensional HLLC Riemann solver for conservation laws: Application to Euler and magnetohydrodynamic flows. J. Comput. Phys. (to appear)
6.
Barichello L.B., Siewert C.E.: A discrete-ordinates solution for a non-grey model with complete frequency redistribution. JQSRT 62, 665–675 (1999)CrossRef
7.
Bechouche P., Gosse L.: A Semiclassical Coupled Model for the Transient Simulation of Semiconductor Devices. SIAM J. Scient. Comput. 29, 376–396 (2007)CrossRefMATHMathSciNet
8.
Berger M.J., Helzel C., LeVeque R.J.: h-box methods for the approximation of conservation laws on irregular grids. SIAM J. Numer. Anal. 41, 893–918 (2003)
9.
10.
Bouchut F.: Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources. Frontiers in Mathematics series. Birkhäuser Verlag, Basel (2004)CrossRefMATH
11.
Bourdarias C.: Convergence of fluctuation-splitting schemes for two dimensional scalar conservation laws with a kinetic solver. Numer. Math. 87, 645–662 (2001)CrossRefMATHMathSciNet
12.
Carrillo J.A., Gamba I., Shu C.W.: Computational macroscopic approximations to the one-dimensional relaxation-time kinetic system for semiconductors. Physica D 2551, 1–18 (2000)MATHMathSciNet
13.
Carrillo J.A., Gamba I., Majorana A., Shu C.W.: A WENO-solver for the transients of Boltzmann-Poisson for semiconductor devices. Performance and comparisons with Monte Carlo methods. J. Comput. Phys. 184, 498–525, 2003MATH
14.
Cercignani C., Gamba I., Levermore C.D.: A drift-collision balance for a Boltzmann-Poisson system in bounded domains. SIAM J. Appl. Math. 64, 1932–1958 (2001)CrossRefMATHMathSciNet
15.
Cheng Y., Gamba I.M., Proft J.: Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations. Math. Comput. 81, 153–190 (2012)CrossRefMATHMathSciNet
16.
Clerc S.: On the preconditioning of finite volumes scheme. In: Hyperbolic problems: theory, numerics, applications, pp. 159–165. International Series of Numerical Mathematics vol. 129. Birkhäuser Verlag, Basel (1999)
17.
18.
19.
Davis S.: A rotationally biased upwind difference scheme for the Euler equations. J. Comput. Phys. 56, 65–92 (1984)CrossRefMATH
20.
De Palma P., Pascazio G., Rubino D.T.: M. Napolitano. Residual distribution schemes for advection and advection-diffusion problems on quadrilateral cells, J. Comput. Phys. 218, 159199 (2006)
21.
Deconinck H., Powell K.G., Roe P.L., Struijs R.: Multi-Dimensional Schemes for Scalar Advection. AIAA paper 90, 65–97 (1991)
22.
Després B., Lagoutiere F.: Genuinely multidimensional non-dissipative finite volume schemes for transport. Intern. J. Applied Math. Comput. Sci. 17, 321–328 (2007)CrossRefMATH
23.
Ding Y., Yang L., Yuan L: A finite difference real ghost fluid method on moving meshes with corner-transport upwind interpolation. Comput. Fluids 49, 247–257 (2011)CrossRefMATHMathSciNet
24.
Eliasson B.: Outflow Boundary Conditions for the Fourier Transformed One-Dimensional Vlasov-Poisson System. J. Scient. Comput. 16, 1–28.
25.
Filbet F.: Convergence of a Finite Volume Scheme for the One Dimensional Vlasov-Poisson System. SIAM J. Numer. Anal. 39, 1146–1169 (2001)CrossRefMATHMathSciNet
26.
Rahmani Firoozjaeea A., Afshar M.H.: Discrete Least Squares Meshless (DLSM) method for simulation of steady state shallow water flows. Scientia Iranica 18, 835–845 (2011)CrossRefMATH
27.
Frosali G., van der Mee C.V.M., Paveri-Fontana S.L.: Conditions for runaway phenomena in the kinetic theory of swarms. J. Math. Phys. 30, 1177–1186 (1989)CrossRefMATHMathSciNet
28.
Gallardo J.M., Parés C., Castro M.: On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas. J. Comput. Phys. 227, 574–601 (2007)CrossRefMATHMathSciNet
29.
Gosse L.: A well-balanced flux splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comput. & Math. Applic. 39, 135–159 (2000)CrossRefMATHMathSciNet
30.
Helzel C., Berger M.J., LeVeque R.J.: A high-resolution rotated grid method for conservation laws with embedded geometries. SIAM J. Sci. Comput. 26, 785–809 (2005)CrossRefMATHMathSciNet
31.
Jüngel A., Transport Equations for Semiconductors. Springer, Berlin Heidelberg (2009)CrossRef
32.
Koren B., Low-diffusion rotated upwind schemes, multigrid and defect correction for steady, multi-dimensional Euler flows. Report NM-R 9021. CWI, Amsterdam (1990)
33.
Kurganov A., Petrova G.: A Second-Order Well-Balanced Positivity Preserving Central-Upwind Scheme for the Saint-Venant System. Comm. Math. Sci. 5, 133–160 (2007)CrossRefMATHMathSciNet
34.
Lavery J.: Solution of steady-state, two-dimensional conservation laws by mathematical programming. SIAM J. Numer. Anal. 28, 141–155 (1991)CrossRefMATHMathSciNet
35.
Levy D.W., Powell K.G., Van Leer B.: Use of rotated Riemann solver for the two-dimensional Euler equations. J. Comput. Phys. 106, 201–214 (1993)CrossRefMATH
36.
Morton K.W., Rudgyard M.A., Shaw G.J.: Upwind Iteration Methods for the Cell Vertex Scheme in One Dimension. J. Comput. Phys. 114, 209–226 (1994)CrossRefMATHMathSciNet
37.
Murillo J., García-Navarro P., Burguete J., Brufau P.: The influence of source terms on stability, accuracy and conservation in two-dimensional shallow flow simulation using triangular finite volumes. Intern. J. Numer. Meth. Fluids 54, 543–590 (2007)CrossRefMATHMathSciNet
38.
Noelle S., Pankratz N., Puppo G., Natvig J.R.: Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213, 474–499 (2006)CrossRefMATHMathSciNet
39.
Paveri-Fontana S.L., van der Mee C.V.M., Zweifel P.F.: A Neutral Gas Model for Electron Swarms. J. Stat. Phys. 83, 247–265 (1999)MathSciNet
40.
Rauch J.: BV Estimates Fail for Most Quasilinear Hyperbolic Systems in Dimensions Greater Than One. Comm. Math. Phys. 106, 481–484 (1986)CrossRefMATHMathSciNet
41.
42.
Roe P.L., Sidilkover D.: Optimum positive linear schemes for advection in 2 and 3 dimensions. SIAM J. Numer. Anal. 29, 1542–1568 (1992)CrossRefMATHMathSciNet
43.
Rondoni L., Zweifel P.: Solutions of singular integral equations from gas dynamics and plasma physics. J. Stat. Phys. 70, 1297–1312 (1993)CrossRefMATHMathSciNet
44.
Rosencher E.: A Displaced Maxwellian approach to ballistic electron transport in semiconductors. Solid State Commun. 38, 1293–1295 (1981)CrossRef
45.
Shu C.-W.: High Order Weighted Essentially Non-oscillatory Schemes for Convection Dominated Problems. SIAM Rev. 51, 82–126 (2009)CrossRefMATHMathSciNet
46.
Sidilkover D., Roe P.L., Unification of Some Advection Schemes in Two Dimensions. Tech. report 95–10. NASA Langley Research Center, Hampton, VA (1995)
47.
Tang H., Tang T.: Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws. SIAM J. Numer. Anal. 41, 487–515 (2003)CrossRefMATHMathSciNet
48.
Toepffer C., Cercignani C.: Analytical results for the Boltzmann equation. Contrib. Plasma Phys. 37, 279–291 (1997)CrossRef
49.
Toro E.F., Garcia-Navarro P.: Godunov-type methods for free-surface shallow flows: A review. Journal of Hydraulic Research 45, 736–751 (2007)CrossRef
50.
Zheng Y.: Systems of Conservation Laws: Two-Dimensional Riemann Problems. Birkhäuser, Boston (2001)CrossRefMATH
Metadaten
Titel
Balances in Two Dimensions: Kinetic Semiconductor Equations Again
verfasst von
Laurent Gosse
Copyright-Jahr
2013
Verlag
Springer Milan
Buch
Computing Qualitatively Correct Approximations of Balance Laws

Print ISBN: 978-88-470-2891-3

Electronic ISBN: 978-88-470-2892-0

Copyright-Jahr: 2013

DOI
https://doi.org/10.1007/978-88-470-2892-0_15