Skip to main content
Top
Published in: BIT Numerical Mathematics 2/2015

01-06-2015

A well-balanced and asymptotic-preserving scheme for the one-dimensional linear Dirac equation

Author: Laurent Gosse

Published in: BIT Numerical Mathematics | Issue 2/2015

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

The numerical approximation of one-dimensional relativistic Dirac wave equations is considered within the recent framework consisting in deriving local scattering matrices at each interface of the uniform Cartesian computational grid. For a Courant number equal to unity, it is rigorously shown that such a discretization preserves exactly the \(L^2\) norm despite being explicit in time. This construction is well-suited for particles for which the reference velocity is of the order of \(c\), the speed of light. Moreover, when \(c\) diverges, that is to say, for slow particles (the characteristic scale of the motion is non-relativistic), Dirac equations are naturally written so as to let a “diffusive limit” emerge numerically, like for discrete 2-velocity kinetic models. It is shown that an asymptotic-preserving scheme can be deduced from the aforementioned well-balanced one, with the following properties: it yields unconditionally a classical Schrödinger equation for free particles, but it handles the more intricate case with an external potential only conditionally (the grid should be such that \(c \Delta x\rightarrow 0\)). Such a stringent restriction on the computational grid can be circumvented easily in order to derive a seemingly original Schrödinger scheme still containing tiny relativistic features. Numerical tests (on both linear and nonlinear equations) are displayed.

Dont have a licence yet? Then find out more about our products and how to get one now:

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!

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!

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!

Appendix
Available only for authorised users
Literature
1.
go back to reference Alvarez, A.: Linearized Crank–Nicolson scheme for nonlinear Dirac equations. J. Comp. Phys. 99, 348–350 (1992)CrossRefMATH Alvarez, A.: Linearized Crank–Nicolson scheme for nonlinear Dirac equations. J. Comp. Phys. 99, 348–350 (1992)CrossRefMATH
2.
go back to reference Alvarez, A., Kuo, P.-Y., Vazquez, L.: The numerical study of a nonlinear one-dimensional Dirac equation. Appl. Math. Comput. 13, 1–15 (1983)CrossRefMATHMathSciNet Alvarez, A., Kuo, P.-Y., Vazquez, L.: The numerical study of a nonlinear one-dimensional Dirac equation. Appl. Math. Comput. 13, 1–15 (1983)CrossRefMATHMathSciNet
3.
go back to reference Askar, A., Cakmak, A.S.: Explicit integration method for the time dependent Schrödinger equation for collision problems. J. Chem. Phys. 68, 2794–2798 (1978)CrossRefMathSciNet Askar, A., Cakmak, A.S.: Explicit integration method for the time dependent Schrödinger equation for collision problems. J. Chem. Phys. 68, 2794–2798 (1978)CrossRefMathSciNet
5.
go back to reference Bechouche, P., Mauser, N., Poupaud, F.: Semi-(non)relativistic limits of the Dirac equation with external time-dependent electromagnetic fields. Commun. Math Phys. 197, 405–425 (1998)CrossRefMATHMathSciNet Bechouche, P., Mauser, N., Poupaud, F.: Semi-(non)relativistic limits of the Dirac equation with external time-dependent electromagnetic fields. Commun. Math Phys. 197, 405–425 (1998)CrossRefMATHMathSciNet
6.
go back to reference Berthon, C., Sarazin, C., Turpault, R.: Space-time generalized Riemann problem solvers of order k for linear advection with unrestricted time step. J. Sci. Comput. 55, 268–308 (2013)CrossRefMATHMathSciNet Berthon, C., Sarazin, C., Turpault, R.: Space-time generalized Riemann problem solvers of order k for linear advection with unrestricted time step. J. Sci. Comput. 55, 268–308 (2013)CrossRefMATHMathSciNet
7.
8.
go back to reference Bournaveas, N., Zouraris, G.E.: Theory and numerical approximations for a nonlinear Dirac system. Math. Model. Numer. Anal. (M2AN) 46, 841–874 (2012)CrossRefMATHMathSciNet Bournaveas, N., Zouraris, G.E.: Theory and numerical approximations for a nonlinear Dirac system. Math. Model. Numer. Anal. (M2AN) 46, 841–874 (2012)CrossRefMATHMathSciNet
9.
go back to reference Carles, R., Mohammadi, B.: Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime. Math. Model. Numer. Anal. (M2AN) 45, 981–1008 (2011)CrossRefMATHMathSciNet Carles, R., Mohammadi, B.: Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime. Math. Model. Numer. Anal. (M2AN) 45, 981–1008 (2011)CrossRefMATHMathSciNet
10.
11.
go back to reference Jing-Bo, Chen, Hong, Liu: Two kinds of square-conservative integrators for nonlinear evolution equations. Chin. Phys. Lett. 25, 1168–1171 (2008)CrossRef Jing-Bo, Chen, Hong, Liu: Two kinds of square-conservative integrators for nonlinear evolution equations. Chin. Phys. Lett. 25, 1168–1171 (2008)CrossRef
12.
go back to reference Cotaescu, I., Gravila, P., Paulescu, M.: Applying the Dirac equation to derive the transfer matrix for piecewise constant potentials. Phys. Lett. A 366(4), 363–366 (2007)CrossRef Cotaescu, I., Gravila, P., Paulescu, M.: Applying the Dirac equation to derive the transfer matrix for piecewise constant potentials. Phys. Lett. A 366(4), 363–366 (2007)CrossRef
13.
go back to reference Degond, P., Gallego, S., Méhats, F.: An asymptotic preserving scheme for the Schrödinger equation in the semiclassical limit. C. R. Math. Acad. Sci. Paris 345, 531–536 (2007)CrossRefMATHMathSciNet Degond, P., Gallego, S., Méhats, F.: An asymptotic preserving scheme for the Schrödinger equation in the semiclassical limit. C. R. Math. Acad. Sci. Paris 345, 531–536 (2007)CrossRefMATHMathSciNet
14.
go back to reference Domingez-Adame, F., Gonzalez, M.A.: Solvable linear potentials in the Dirac equation. Europhys. Lett. 13(3), 193–198 (1990)CrossRef Domingez-Adame, F., Gonzalez, M.A.: Solvable linear potentials in the Dirac equation. Europhys. Lett. 13(3), 193–198 (1990)CrossRef
15.
go back to reference Domingez-Adame, F., Rodriguez, A.: A one-dimensional relativistic screened Coulomb potential. Phys. Lett. A 198, 275–278 (1995)CrossRef Domingez-Adame, F., Rodriguez, A.: A one-dimensional relativistic screened Coulomb potential. Phys. Lett. A 198, 275–278 (1995)CrossRef
16.
go back to reference Duchêne, V., Marzuola, J.L., Weinstein, M.I.: Wave operator bounds for one-dimensional Schrödinger operators with singular potentials and applications. J. Math. Phys. 52, 013505 (2011)CrossRefMathSciNet Duchêne, V., Marzuola, J.L., Weinstein, M.I.: Wave operator bounds for one-dimensional Schrödinger operators with singular potentials and applications. J. Math. Phys. 52, 013505 (2011)CrossRefMathSciNet
17.
go back to reference De Frutos, J., Sanz-Serna, J.M.: Split-step spectral schemes for nonlinear Dirac systems. J. Comp. Phys. 83, 407–423 (1989)CrossRefMATH De Frutos, J., Sanz-Serna, J.M.: Split-step spectral schemes for nonlinear Dirac systems. J. Comp. Phys. 83, 407–423 (1989)CrossRefMATH
18.
go back to reference Gosse, L.: Computing Qualitatively Correct Approximations of Balance Laws. Springer ISBN 978-88-470-2891-3 (2013) Gosse, L.: Computing Qualitatively Correct Approximations of Balance Laws. Springer ISBN 978-88-470-2891-3 (2013)
19.
go back to reference Gosse, L.: MUSCL reconstruction and Haar wavelets. Commun. Math. Sci. (submitted) (2014) Gosse, L.: MUSCL reconstruction and Haar wavelets. Commun. Math. Sci. (submitted) (2014)
20.
go back to reference Gosse, L., Toscani, G.: An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations. C. R. Math. Acad. Sci. Paris 334, 337–342 (2002)CrossRefMATHMathSciNet Gosse, L., Toscani, G.: An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations. C. R. Math. Acad. Sci. Paris 334, 337–342 (2002)CrossRefMATHMathSciNet
21.
go back to reference Guo, B.Z., Zwart, H.: On the relation between stability of continuous- and discrete-time evolution equations via the Cayley transform. Integr. Equ. Oper. Theory 54, 349–383 (2006)CrossRefMATHMathSciNet Guo, B.Z., Zwart, H.: On the relation between stability of continuous- and discrete-time evolution equations via the Cayley transform. Integr. Equ. Oper. Theory 54, 349–383 (2006)CrossRefMATHMathSciNet
22.
go back to reference Hairer, E., Lubich, C.H., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics, vol. 31. Springer, Berlin, Heidelberg (2006) Hairer, E., Lubich, C.H., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics, vol. 31. Springer, Berlin, Heidelberg (2006)
23.
go back to reference Hiller, J.: Solution of the one-dimensional Dirac equation with a linear scalar potential. Am. J. Phys. 70(5), 522–524 (2002)CrossRef Hiller, J.: Solution of the one-dimensional Dirac equation with a linear scalar potential. Am. J. Phys. 70(5), 522–524 (2002)CrossRef
24.
go back to reference Hong, Jialin, Li, Chun: Multi-symplectic Runge–Kutta methods for nonlinear Dirac equations. J. Comput. Phys. 211, 448–472 (2006)CrossRefMATHMathSciNet Hong, Jialin, Li, Chun: Multi-symplectic Runge–Kutta methods for nonlinear Dirac equations. J. Comput. Phys. 211, 448–472 (2006)CrossRefMATHMathSciNet
25.
go back to reference de la Hoz, F., Vadillo, F.: An integrating factor for nonlinear Dirac equations. Comput. Phys. Commun. 181, 1195–1203 (2010)CrossRefMATH de la Hoz, F., Vadillo, F.: An integrating factor for nonlinear Dirac equations. Comput. Phys. Commun. 181, 1195–1203 (2010)CrossRefMATH
26.
go back to reference Huang, Zhongyi, Jin, Shi, Markowich, Peter A., Sparber, Christof, Zheng, Chunxiong: A time-splitting spectral scheme for the Maxwell–Dirac system. J. Comput. Phys. 208, 761–789 (2005)CrossRefMATHMathSciNet Huang, Zhongyi, Jin, Shi, Markowich, Peter A., Sparber, Christof, Zheng, Chunxiong: A time-splitting spectral scheme for the Maxwell–Dirac system. J. Comput. Phys. 208, 761–789 (2005)CrossRefMATHMathSciNet
27.
28.
go back to reference Jin, S.: Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21, 441–454 (1999)CrossRefMATHMathSciNet Jin, S.: Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21, 441–454 (1999)CrossRefMATHMathSciNet
29.
go back to reference Jin, S., Markowich, P.A., Sparber, C.: Mathematical and computational methods for semiclassical Schrodinger equations. Acta Numer. 20, 211–289 (2011)CrossRefMathSciNet Jin, S., Markowich, P.A., Sparber, C.: Mathematical and computational methods for semiclassical Schrodinger equations. Acta Numer. 20, 211–289 (2011)CrossRefMathSciNet
30.
go back to reference Kong, Linghua, Liu, Ruxun, Zheng, Xiaohong: A survey on symplectic and multi-symplectic algorithms. Appl. Math. Comput. 186, 670–684 (2007)CrossRefMATHMathSciNet Kong, Linghua, Liu, Ruxun, Zheng, Xiaohong: A survey on symplectic and multi-symplectic algorithms. Appl. Math. Comput. 186, 670–684 (2007)CrossRefMATHMathSciNet
31.
go back to reference LeFloch, Ph, Tzavaras, A.E.: Representation of weak limits and definition of nonconservative products. SIAM J. Math. Anal. 30, 1309–1342 (1999)CrossRefMATHMathSciNet LeFloch, Ph, Tzavaras, A.E.: Representation of weak limits and definition of nonconservative products. SIAM J. Math. Anal. 30, 1309–1342 (1999)CrossRefMATHMathSciNet
32.
go back to reference Lubich, C: Integrators for quantum dynamics: a numerical analyst’s brief review. In: Grotendorst, J., Marx, D., Muramatsu, A. (Eds.) Quantum simulations of complex many-body systems: from theory to algorithms, pp. 459–466. John von Neumann Institute for Computing, Jülich, NIC Series 10, ISBN 3-00-009057-6 (2002) Lubich, C: Integrators for quantum dynamics: a numerical analyst’s brief review. In: Grotendorst, J., Marx, D., Muramatsu, A. (Eds.) Quantum simulations of complex many-body systems: from theory to algorithms, pp. 459–466. John von Neumann Institute for Computing, Jülich, NIC Series 10, ISBN 3-00-009057-6 (2002)
33.
go back to reference Markowich, P.A., Pietra, P., Pohl, C.: Numerical approximation of quadratic observables of Schrödinger-type equations in the semiclassical limit. Numer. Math. 81, 595–630 (1999)CrossRefMATHMathSciNet Markowich, P.A., Pietra, P., Pohl, C.: Numerical approximation of quadratic observables of Schrödinger-type equations in the semiclassical limit. Numer. Math. 81, 595–630 (1999)CrossRefMATHMathSciNet
35.
go back to reference Micu, S.: Uniform boundary controllability of a semi-discrete 1-D wave equation with vanishing viscosity. SIAM J. Control Optim. 47, 2857–2885 (2008)CrossRefMATHMathSciNet Micu, S.: Uniform boundary controllability of a semi-discrete 1-D wave equation with vanishing viscosity. SIAM J. Control Optim. 47, 2857–2885 (2008)CrossRefMATHMathSciNet
36.
go back to reference Morsink, S.M., Mann, R.B.: Black hole radiation of Dirac particles in 1+1 dimensions. Class. Quantum Grav. 8, 2257 (1991)CrossRefMathSciNet Morsink, S.M., Mann, R.B.: Black hole radiation of Dirac particles in 1+1 dimensions. Class. Quantum Grav. 8, 2257 (1991)CrossRefMathSciNet
37.
go back to reference Noelle, S.: Hyperbolic systems of conservation laws, the Weyl equation, and multidimensional upwinding. J. Comput. Phys. 115, 22–26 (1994)CrossRefMATHMathSciNet Noelle, S.: Hyperbolic systems of conservation laws, the Weyl equation, and multidimensional upwinding. J. Comput. Phys. 115, 22–26 (1994)CrossRefMATHMathSciNet
38.
go back to reference Sinha, A., Roychoudury, R.: Dirac equation in (1+1)-dimensional curved space-time. Int. J. Theor. Phys. 33, 1511–1522 (1994)CrossRefMATH Sinha, A., Roychoudury, R.: Dirac equation in (1+1)-dimensional curved space-time. Int. J. Theor. Phys. 33, 1511–1522 (1994)CrossRefMATH
39.
go back to reference Solomon, D.: An exact solution of the Dirac equation for a time-dependent Hamiltonian in 1\(-\)1 dimension space-time. Can. J. Phys. 88, 137–138 (2010)CrossRef Solomon, D.: An exact solution of the Dirac equation for a time-dependent Hamiltonian in 1\(-\)1 dimension space-time. Can. J. Phys. 88, 137–138 (2010)CrossRef
41.
go back to reference Succi, S.: Numerical solution of the Schrödinger equation using discrete kinetic theory. Phys. Rev. E 53, 1969–1975 (1996)CrossRef Succi, S.: Numerical solution of the Schrödinger equation using discrete kinetic theory. Phys. Rev. E 53, 1969–1975 (1996)CrossRef
43.
go back to reference Thaller, B.: Advanced Visual Quantum Mechanics, (Chapter 7). Springer, New York ISBN 0-387-20777-5 (2005) Thaller, B.: Advanced Visual Quantum Mechanics, (Chapter 7). Springer, New York ISBN 0-387-20777-5 (2005)
44.
go back to reference Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 3rd edn, pp. 427–429. Springer, Berlin, Heidelberg (2009). doi:10.1007/b79761 Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 3rd edn, pp. 427–429. Springer, Berlin, Heidelberg (2009). doi:10.​1007/​b79761
45.
go back to reference Weinberger, P.: All you need to know about the Dirac equation. Philos. Mag. 88(18–20), 2585–2601 (2008)CrossRef Weinberger, P.: All you need to know about the Dirac equation. Philos. Mag. 88(18–20), 2585–2601 (2008)CrossRef
46.
go back to reference Wessels, P.P.F., Caspers, W.J., Wiegel, F.W.: Discretizing the one-dimensional Dirac equation. Europhys. Lett. 46(2), 123–126 (1999)CrossRef Wessels, P.P.F., Caspers, W.J., Wiegel, F.W.: Discretizing the one-dimensional Dirac equation. Europhys. Lett. 46(2), 123–126 (1999)CrossRef
47.
48.
go back to reference Zhang, Y.: Global strong solution to a nonlinear Dirac-type equation in one dimension. Nonlinear Anal.: Theory, Methods Appl. 80, 150–155 (2013)CrossRefMATH Zhang, Y.: Global strong solution to a nonlinear Dirac-type equation in one dimension. Nonlinear Anal.: Theory, Methods Appl. 80, 150–155 (2013)CrossRefMATH
50.
go back to reference Zuazua, E.: Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47, 197–243 (2005)CrossRefMATHMathSciNet Zuazua, E.: Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47, 197–243 (2005)CrossRefMATHMathSciNet
Metadata
Title
A well-balanced and asymptotic-preserving scheme for the one-dimensional linear Dirac equation
Author
Laurent Gosse
Publication date
01-06-2015
Publisher
Springer Netherlands
Published in
BIT Numerical Mathematics / Issue 2/2015
Print ISSN: 0006-3835
Electronic ISSN: 1572-9125
DOI
https://doi.org/10.1007/s10543-014-0510-4

Other articles of this Issue 2/2015

BIT Numerical Mathematics 2/2015 Go to the issue

Premium Partner