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:
Buchtitelbild

2013 | OriginalPaper | Buchkapitel

1. Introduction and Chronological Perspective

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

This introductory chapter aims at positioning the book’s primary topics according to both a scientific and an historic context; loosely speaking, the objective here is more trying to unify seemingly different sectors in numerical analysis rather than being very specific (this will come later on). In particular, one can figure out the main ideas exposed in the sequel by examining very classical computations which trace back to 1960–70, namely the passage from finite differences to exponentially-fitted schemes for transient convection-diffusion equations. In some sense, well-balanced schemes are but an extension of these methods for hyperbolic problems: the link being provided by both the finite volumes discretization (what was formerly called the “box scheme”) and the exact solving of the steady-state equations in order to compute the numerical fluxes at each interface of the computational grid. A point of crucial importance is the following (quoting [61, p. 159]):

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
Nächstes Kapitel Lifting a Non-Resonant Scalar Balance Law
Fußnoten
1
A both WB/AP scheme is necessary as soon as the small parameter varies in x.
 
2
See also the Asymptotic Integration Method, related to these matters.
 
Literatur
1.
de G. Allen D.N.: A suggested approach to finite-difference representation of differential equations, with an application to determine temperature-distributions near a sliding contact. Quart. J. Mech. Appl. Math. 15, 11–33 (1962)
2.
de G. Allen D.N., Southwell R.: Relaxation methods applied to determining the motion, in two dimensions, of a viscous fluid past a fixed cylinder. Quart. J. Mech. Appl. Math. 8, 129–145 (1955)
3.
Arora M., Roe P.L.: On postshock oscillations due to capturing schemes in unsteady flows. J. Comput. Phys. 130, 25–40 (1997)CrossRefMATHMathSciNet
4.
Barichello L.B., Siewert C.E.: A new version of the discrete-ordinates method, Proc. of the 2nd Conf. on Comput. Heat and Mass Transfer (COPPE/EE/UFRJ), October 22–26 2001
5.
Barrett K.E.: The numerical solution of singular-perturbation boundary-value problems. Quart. J. Mech. Appl. Math. 13, 487–507 (1960)CrossRefMathSciNet
6.
7.
Ben Abdallah N., Dolbeault J.: Relative entropies for the Vlasov-Poisson system in bounded domains C. R. Acad. Sci. Paris Série I Math. 330, 867–872 (2000)CrossRefMATHMathSciNet
8.
Ben Abdallah N., Gamba I., Klar A.: The Milne problem for high field kinetic equations. SIAM J. Applied Math. 64, 1739–1736 (2004)MATHMathSciNet
9.
Ben-Artzi M., Falcovitz J.: Generalized Riemann problems in computational fluid dynamics, Cambridge Monographs on Applied and Computational Mathematics, vol. 11 (2003)
10.
11.
Bianchini S., Bressan A.: Vanishing viscosity solutions of nonlinear hyperbolic systems. Annals of Mathematics, 161, 223–242 (2005)CrossRefMATHMathSciNet
12.
Bouchut F., Perthame B.: Kružkov’s inequalities for scalar conservation laws revisited. Trans. Amer. Math. Soc. 350, 2847–2870 (1998)CrossRefMATHMathSciNet
13.
Bremer Y.: Order preserving vibrating strings and applications to Electrodynamics and Mag- netohydrodynamics. Meth. Applic. Anal. 11, 515–532 (2004)
14.
Bremer Y., Corrias L.: A kinetic formulation for multibranch entropy solutions of scalar conservation laws. Ann. I.H.P. Nonlinear Anal. 15, 169–190 (1998)
15.
Brezzi F., Marini L.D., Pietra P.: Two-dimensional exponential fitting and application to driftdiffusion models. SIAM J. Numer. Anal. 26, 1342–1355 (1989)CrossRefMATHMathSciNet
16.
Burschka M.A., Titulaer U.M.: The Kinetic Boundary Layer for the Fokker-Planck Equation with Absorbing Boundary. J. Stat. Phys. 25, 569–582 (1981)CrossRefMathSciNet
17.
18.
Cercignani C.: Elementary solutions of the linearized gas-dynamics Boltzmann equation and their application to the slip-flow problem. Ann. Physics 20, 219–233 (1962)CrossRefMATHMathSciNet
19.
Chen G.-Q., Levermore C.D., Liu T.P.: Hyperbolic conservation laws with stiff relaxation terms and entropy. Comm. Pure Applied Math. 47, 787–830 (1994)CrossRefMATHMathSciNet
20.
Crank J., Nicolson P.: A practical method for numerical evalutation of solutions of partial differential equations of the heat-conduction type. Proc. Cambridge Philos. Soc. 43, 50–67 (1947)CrossRefMATHMathSciNet
21.
22.
Degond P., Guyot-Delaurens F.: Particle Simulations of the Semiconductor Boltzmann Equation for One Dimensional Inhomogeneous Structures. J. Comput. Phys. 90, 65–97 (1990)CrossRefMATHMathSciNet
23.
Dennis S.C.R.: Finite differences associated with second-order differential equations. Quart. J. Mech. Appl. Math. 13, 487–507 (1960)CrossRefMATHMathSciNet
24.
Descombes S., Dia B.O.: An Operator-Theoretic Proof of an Estimate on the Transfer Operator. J. Funct. Anal. 165, 240–257 (1999)CrossRefMATHMathSciNet
25.
Efraimsson G., Kreiss G.: A note on the effect of artificial viscosity on solutions of conservation laws. Apl. Numer. Math. 21, 155–173 (1996)CrossRefMATHMathSciNet
26.
Engquist B., Sjögreen B.: The Convergence Rate of Finite Difference Schemes in the Presence of Shocks. SIAM J. Numer. Anal. 35, 2464–2485 (1998)CrossRefMATHMathSciNet
27.
Eymard R., Fuhrmann J., Gärtner K.: A finite volume scheme for parabolic equations derived from one-dimensional local Dirichlet problems. Numer. Math. 102, 463–495 (2006)CrossRefMATHMathSciNet
28.
Faou E.: Geometric numerical integration and Schrödinger equations. Zürich lectures in advanced mathematics, European Math. Soc. (2012)
29.
Filbet F.: Convergence of a Finite Volume Scheme for the One Dimensional Vlasov-Poisson System. SIAM J. Numer. Anal. 39, 1146–1169 (2001)CrossRefMATHMathSciNet
30.
31.
Ford R.M., Cummings P.T.: On the relationship between cell balance equations for chemotactic cell populations. SIAM J. Applied Math. 52, 1426–1441 (1992)CrossRefMATHMathSciNet
32.
33.
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
34.
Gartland Jr. E.C.: On the uniform convergence of the Scharfetter-Gummel discretization in one dimension. SIAM J. Numer. Anal. 30, 749–758 (1993)CrossRefMATHMathSciNet
35.
Glimm J., Marshall G., Plohr B.J.: A generalized Riemann problem for quasi one dimensional gas flows. Adv. Appl. Math. 5, 1–30 (1984)CrossRefMATHMathSciNet
36.
Gosse L.: Well-balanced schemes using elementary solutions for linear models of the Boltzmann equation in one space dimension. Kinetic and Related Models 5, 283–323 (2012)CrossRefMATHMathSciNet
37.
Greenberg J., Alt W.: Stability results for a diffusion equation with functional shift approximating achemotaxis model. Trans. Amer. Math. Soc. 300, 235–258 (1987)CrossRefMATHMathSciNet
38.
Greenberg J., LeRoux A.Y.: A well balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33, 1–16 (1996)CrossRefMathSciNet
39.
Guerra G.: Well-posedness for a scalar conservation law with singular nonconservative source. J. Diff. Eqns. 206, 438–469 (2004)CrossRefMATHMathSciNet
40.
41.
Hairer E., Lubich Ch., Wanner G.: Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, Springer Series in Computational Mathematics. Springer-Verlag, Berlin Heidelberg (2006)MATH
42.
Il’in A.M.: Differencing scheme for a differential equation with a small parameter affecting the highest derivative. Mat. Zamtki (1969) 6, 237–248 (1947)
43.
Isaacson E., Temple B.: Convergence of the 2 × 2 Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math. 55, 625–640 (1995)CrossRefMATHMathSciNet
44.
Jenny P., Müller B.: Rankine-Hugoniot Riemann solver considering source terms and multidimensional effects. J. Comp. Phys. 145, 575–610 (1998)CrossRefMATHMathSciNet
45.
Jerome J.W.: Drift-diffusion systems: variational principles and fixed point maps for steady state semiconductor models. In: Hess K., Leburton J.P., Ravaioli U. (eds.) Computational Electronics, Semiconductor Transport and Device, pp. 15–20. Kluwer (1991)
46.
47.
Jin S.: Efficient Asymptotic-Preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Scient. Comput. 21, 441–454 (2000)CrossRefMATHMathSciNet
48.
Jin S., Levermore C.D.: Numerical schemes for hyperbolic conservation laws with stiff relaxation terms. J. Comp. Phys. 126, 449–467 (1996)CrossRefMATHMathSciNet
49.
Kainz A.J., Titulaer U.M.: The structure of the stationary kinetic boundary layer for the linear BGK equation. J. Phys. A: Math. Gen. 25, 3189–3203 (1992)CrossRefMATHMathSciNet
50.
51.
52.
Kopteva N.V.: On the Uniform in Small Parameter Convergence of a Weighted Scheme for the One-Dimensional Time-Dependent Convection-Diffusion Equation. Comput. Math. & Math. Phys. 37, 1173–1180 (1997)MATHMathSciNet
53.
Kriese J.T., Chang T.S., Siewert C.E.: Elementary solutions of coupled model equations in the kinetic theory of gases. Int. J. Eng. Sci. 12, 441–470 (1974)CrossRefMATHMathSciNet
54.
Kuo C.C.J., Levy B.: Mode-dependent finite-difference discretization of linear homogeneous differential equations, SIAM J. Scient. Comput. 9, 992–1015 (1988)CrossRefMATHMathSciNet
55.
LeFloch P.G., Raviart P.A.: An asymptotic expansion for the solution of the generalized Riemann problem. Part I: general theory. Ann. I.H.P. Nonlinear Analysis 5, 179–205 (1989)
56.
Levermore C.D.: Personal Communication, October 2004
57.
58.
59.
60.
Marshall G., Menéndez A.N.: Numerical treatment of nonconservation forms of the equations of shallow water theory. J. Comp. Phys. 44, 167–188 (1981)CrossRefMATH
61.
Oran E.S., Boris J.P.: Numerical Simulation of Reactive Flow, 2nd edn. Cambridge Univ. Press, Cambridge (2005)MATH
62.
Othmer H., Hillen T.: The diffusion limit of transport equations II: Chemotaxis equations. SIAM J. Appl. Math. 62, 1222–1250, (2002)CrossRefMATHMathSciNet
63.
Pagani C.D.: Studio di alcune questioni concernenti l’equazione generalizzata di Fokker- Planck. Boll. Un. Mat. Ital. 3(4), 961–986 (1970)MATHMathSciNet
64.
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
65.
Roe P.L.: Upwind differencing schemes for hyperbolic conservation laws with source terms in Nonlinear Hyperboüc Problems. In: Carasso C., Raviart P.-A., Serre D. (eds.), Lecture Notes in Mathematics, vol. 1270, pp. 41–55. Springer-Verlag, Berlin Heidelberg (1986)
66.
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
67.
Rondoni L., Zweifel P.: Solutions of singular integral equations from gas dynamics and plasma physics. J. Stat. Phys. 70, 1297–1312 (1993)CrossRefMATHMathSciNet
68.
Roos H.-G., Stynes M., Tobiska L.: Robust Numerical Methods for Singularly Perturbed Differential Equations, Convection-Diffusion-reaction and Flow Problems, 2nd edn. Springer Series in Computational Mathematics, vol. 24. Springer-Verlag, Berlin Heidelberg (2008)
69.
Saito N.: An interpretation of the Scharfetter-Gummel finite difference scheme. Proc. Japan Acad. Ser. A Math. Sci. 82, 187–191 (2006)CrossRefMATHMathSciNet
70.
Scharfetter H.L., Gummel H.K.: Large signal analysis of a silicon Read diode oscillator. IEEE Trans. Electron Devices 16, 64–77 (1969)CrossRef
71.
Sethian J.A.: Level Set Methods: Evolving Interfaces in Geometry. Fluid Mechanics, Computer Vision and Materials Sciences, Cambridge University Press, Cambridge (1996)MATH
72.
Siklosi M., Batzorig B., Kreiss G.: An investigation of the internal structure of shock profiles for shock capturing schemes. J. Comput. Appl. Math. 201, 8–29 (2007)CrossRefMATHMathSciNet
73.
Sod G.A.: A numerical study of oxygen diffusion in a spherical cell with the Michaelis-Menten oxygen uptake kinetics. J. Math. Biol. 24, 279–289 (1986)CrossRefMATH
74.
Stakgold I.: Green’s functions and boundary value problems. John Wiley & Sons Inc., Hoboken (1979)MATH
75.
Su B., Olson G.L.: An Analytical Benchmark for Non-Equilibrium Radiative Transfer in an Isotropically Scattering Medium. Ann. Nucl. Energy 24, 1035–1055 (1997)CrossRef
76.
ten Thije Boonkkamp J.H.M., Anthonissen M.J.H.: The Finite Volume-Complete Flux Scheme for Advection-diffusion-Reaction Equations. J. Sci. Comput. 46, 47–70 (2011)CrossRefMATHMathSciNet
77.
Toepffer C., Cercignani C.: Analytical results for the Boltzmann equation. Contrib. Plasma Phys. 37, 279–291 (1997)CrossRef
78.
Vasilieva A.B.: On the development of singular perturbation theory at Moscow State University and elsewhere. SIAM Review 36, 440–452 (1994)CrossRefMATHMathSciNet
79.
Veling E.J.M.: Asymptotic analysis of a singular Sturm-Liouville boundary value problem, Integral Equations and Operator Theory 7, 561–587 (1984)CrossRefMATHMathSciNet
80.
Weinan E., Homogeneization of scalar conservation laws with oscillatory forcing terms. SIAM J. Appl. Math. 52, 959–972 (1992)CrossRefMATHMathSciNet
Metadaten
Titel
Introduction and Chronological Perspective
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_1