A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation. (English) Zbl 1433.65234
Summary: We propose a new fourth-order compact time-splitting (\(S_\mathrm{4c}\)) Fourier pseudospectral method for the Dirac equation by splitting the Dirac equation into two parts together with using the double commutator between them to integrate the Dirac equation at each time interval. The method is explicit, fourth-order in time and spectral order in space. It is unconditionally stable and conserves the total probability in the discretized level. It is called a compact time-splitting method since, at each time step, the number of substeps in \(S_\mathrm{4c}\) is much less than those of the standard fourth-order splitting method and the fourth-order partitioned Runge-Kutta splitting method. Another advantage of \(S_\mathrm{4c}\) is that it avoids to use negative time steps in integrating subproblems at each time interval. Comparison between \(S_\mathrm{4c}\) and many other existing time-splitting methods for the Dirac equation is carried out in terms of accuracy and efficiency as well as longtime behavior. Numerical results demonstrate the advantage in terms of efficiency and accuracy of the proposed \(S_\mathrm{4c}\). Finally, we report the spatial/temporal resolutions of \(S_\mathrm{4c}\) for the Dirac equation in different parameter regimes including the nonrelativistic limit regime, the semiclassical limit regime, and the simultaneously nonrelativistic and massless limit regime.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
Keywords:
Dirac equation; fourth-order compact time-splitting; double commutator; probability conservation; nonrelativistic limit regime; semiclassical limit regimeReferences:
| [1] | Abanin, D.A., Morozov, S.V., Ponomarenko, L.A., Gorbachev, R.V., Mayorov, A.S., Katsnelson, M.I., Watanabe, K., Taniguchi, T., Novoselov, K.S., Levitov, L.S., Geim, A.K.: Giant nonlocality near the Dirac point in graphene. Science 332, 328-330 (2011) · doi:10.1126/science.1199595 |
| [2] | Antoine, X., Bao, W., Besse, C.: Computational methods for the dynamics of the nonlinear Schrodinger/Gross-Pitaevskii equations. Comput. Phys. Commun. 184, 2621-2633 (2013) · Zbl 1344.35130 · doi:10.1016/j.cpc.2013.07.012 |
| [3] | Antoine, X., Lorin, E.: Computational performance of simple and efficient sequential and parallel Dirac equation solvers, Hal-01496817 (2017) · Zbl 1411.35234 |
| [4] | Antoine, X., Lorin, E., Sater, J., Fillion-Gourdeau, F., Bandrauk, A.D.: Absorbing boundary conditions for relativistic quantum mechanics equations. J. Comput. Phys. 277, 268304 (2014) · Zbl 1349.81080 · doi:10.1016/j.jcp.2014.07.037 |
| [5] | Arnold, A., Steinrück, H.: The ‘electromagnetic’ Wigner equation for an electron with spin. ZAMP 40, 793-815 (1989) · Zbl 0701.35130 |
| [6] | Bader, P., Iserles, A., Kropielnicka, K., Singh, P.: Effective approximation for the linear time-dependent Schrödinger equation. Found. Comput. Math. 14, 689-720 (2014) · Zbl 1302.65230 · doi:10.1007/s10208-013-9182-8 |
| [7] | Bao, W., Cai, Y.: Mathematical theory and numerical methods for Bose-Einstein condensation. Kinet. Relat. Mod. 6, 1-135 (2013) · Zbl 1266.82009 |
| [8] | Bao, W., Cai, Y., Jia, X., Tang, Q.: A uniformly accurate multiscale time integrator pseudospectral method for the Dirac equation in the nonrelativistic limit regime. SIAM J. Numer. Anal. 54, 1785-1812 (2016) · Zbl 1346.35171 · doi:10.1137/15M1032375 |
| [9] | Bao, W., Cai, Y., Jia, X., Tang, Q.: Numerical methods and comparison for the Dirac equation in the nonrelativistic limit regime. J. Sci. Comput. 71, 1094-1134 (2017) · Zbl 1370.81053 · doi:10.1007/s10915-016-0333-3 |
| [10] | Bao, W., Cai, Y., Jia, X., Yin, J.: Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime. Sci. China Math. 59, 1461-1494 (2016) · Zbl 1365.65208 · doi:10.1007/s11425-016-0272-y |
| [11] | Bao, W., Cai, Y., Yin, J.: Super-resolution of the time-splitting methods for the Dirac equation in the nonrelativisitic limit regime, preprint (2018) |
| [12] | Bao, W., Jin, S., Markowich, P.A.: On time-splitting spectral approximation for the Schrödinger equation in the semiclassical regime. J. Comput. Phys. 175, 487-524 (2002) · Zbl 1006.65112 · doi:10.1006/jcph.2001.6956 |
| [13] | Bao, W., Jin, S., Markowich, P.A.: Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semi-classical regimes. SIAM J. Sci. Comput. 25, 27-64 (2003) · Zbl 1038.65099 · doi:10.1137/S1064827501393253 |
| [14] | Bao, W., Li, X.: An efficient and stable numerical methods for the Maxwell-Dirac system. J. Compute. Phys. 199, 663-687 (2004) · Zbl 1056.81080 · doi:10.1016/j.jcp.2004.03.003 |
| [15] | Bao, W., Shen, J.: A fourth-order time-splitting Laguerre-Hermite pseudo-spectral method for Bose-Einstein condensates. SIAM J. Sci. Comput. 26, 2010-2028 (2005) · Zbl 1084.35083 · doi:10.1137/030601211 |
| [16] | Bechouche, P., Mauser, N., Poupaud, F.: (Semi)-nonrelativistic limits of the Dirac equaiton with external time-dependent electromagnetic field. Commun. Math. Phys. 197, 405-425 (1998) · Zbl 0926.35124 · doi:10.1007/s002200050457 |
| [17] | Blanesa, S., Moan, P.C.: Practical symplectic partitioned Runge-Kutta and Runge-Kutta-Nyström methods. J. Comput. Appl. Math. 142, 313-330 (2002) · Zbl 1001.65078 · doi:10.1016/S0377-0427(01)00492-7 |
| [18] | Boada, O., Celi, A., Latorre, J.I., Lewenstein, M.: Dirac equation for cold atoms in artificial curved spacetimes. New J. Phys. 13, 035002 (2011) · Zbl 1448.81302 · doi:10.1088/1367-2630/13/3/035002 |
| [19] | Bolte, J., Keppeler, S.: A semiclassical approach to the Dirac equation. Ann. Phys. 274, 125-162 (1999) · Zbl 0940.35173 · doi:10.1006/aphy.1999.5912 |
| [20] | Caliari, M., Ostermann, A., Piazzola, C.: A splitting approach for the magnetic Schrödinger equation. J. Comput. Appl. Math. 316, 74-85 (2017) · Zbl 1373.81195 · doi:10.1016/j.cam.2016.08.041 |
| [21] | Carles, R.: On Fourier time-splitting methods for nonlinear Schrödinger equations in the semi-classical limit. SIAM J. Numer. Anal. 51, 3232-3258 (2013) · Zbl 1290.35238 · doi:10.1137/120892416 |
| [22] | Carles, R., Gallo, C.: On Fourier time-splitting methods for nonlinear Schrödinger equations in the semi-classical limit II. Analytic regularity. Numer. Math. 136, 315-342 (2017) · Zbl 1368.65198 · doi:10.1007/s00211-016-0841-y |
| [23] | Chin, S.A.: Symplectic integrators from composite operator factorizations. Phys. Lett. A 226, 344-348 (1997) · Zbl 0962.65501 · doi:10.1016/S0375-9601(97)00003-0 |
| [24] | Chin, S.A., Chen, C.R.: Fourth order gradient symplectic integrator methods for solving the time-dependent Schrödinger equation. J. Chem. Phys. 114, 7338-7341 (2001) · doi:10.1063/1.1362288 |
| [25] | Chin, S.A., Chen, C.R.: Gradient symplectic algorithms for solving the Schrödinger equation with time-dependent potentials. J. Chem. Phys. 117, 1409-1415 (2002) · doi:10.1063/1.1485725 |
| [26] | Das, A.: General solutions of MaxwellDirac equations in \[1 + 11+1\] dimensional space – time and spatialconfined solution. J. Math. Phys. 34, 3986-3999 (1993) · Zbl 0780.35107 · doi:10.1063/1.530019 |
| [27] | Das, A., Kay, D.: A class of exact plane wave solutions of the MaxwellDirac equations. J. Math. Phys. 30, 2280-2284 (1989) · Zbl 0701.58059 · doi:10.1063/1.528555 |
| [28] | Davydov, A.S.: Quantum Mechanics. Pergamon Press, Oxford (1976) |
| [29] | Dirac, P.A.M.: The quantum theory of the electron. Proc. R. Soc. Lond. A 117, 610-624 (1928) · JFM 54.0973.01 · doi:10.1098/rspa.1928.0023 |
| [30] | Dirac, P.A.M.: A theory of electrons and protons. Proc. R. Soc. Lond. A 126, 360-365 (1930) · JFM 56.0751.02 · doi:10.1098/rspa.1930.0013 |
| [31] | Dirac, P.A.M.: The Principles of Quantum Mechanics, 3rd edn. Oxford University Press, London (1947) · Zbl 0030.04801 |
| [32] | Esteban, M.J., Séré, E.: Existence and multiplicity of solutions for linear and nonlinear Dirac problems. In: Greiner, P.C., Ivrii, V., Seco, L.A., Sulem, C. (eds.) Partial Differential Equations and Their Applications. CRM Proceedings & Lecture Notes, vol. 12, pp. 107-118. American Mathematical Society, Providence, RI (1997) · Zbl 0889.35084 |
| [33] | Fefferman, C.L., Weistein, M.I.: Honeycomb lattice potentials and Dirac points. J. Am. Math. Soc. 25, 11691220 (2012) · Zbl 1316.35214 · doi:10.1090/S0894-0347-2012-00745-0 |
| [34] | Fefferman, C.L., Weistein, M.I.: Wave packets in honeycomb structures and two-dimensional Dirac equations. Commun. Math. Phys. 326, 251-286 (2014) · Zbl 1292.35195 · doi:10.1007/s00220-013-1847-2 |
| [35] | Ferreira, A., Gomes, J.V., Nilsson, J., Mucciolo, E.R., Peres, N.M.R., Catro Neto, A.H.: Unifieddescription of the dc-conductivity of monolayer and bilayer graphene at finite densities based on resonant scatterers. Phys. Rev. B 83, 165402 (2011) · doi:10.1103/PhysRevB.83.165402 |
| [36] | Fillion-Gourdeau, F., Lorin, E., Bandrauk, A.D.: Resonantly enhanced pair production in a simple diatomic model. Phys. Rev. Lett. 110, 013002 (2013) · doi:10.1103/PhysRevLett.110.013002 |
| [37] | Forest, E., Ruth, R.D.: Fourth-order symplectic integration. Phys. D Nonlinear Phenom. 43, 105-117 (1990) · Zbl 0713.65044 · doi:10.1016/0167-2789(90)90019-L |
| [38] | Geng, S.: Syplectic partitioned Runge-Kutta methods. J. Comput. Math. 11, 365-372 (1993) · Zbl 0789.65049 |
| [39] | Gérad, P., Markowich, P.A., Mauser, N.J., Poupaud, F.: Homogenization limits and Wigner transforms. Commun. Pure Appl. Math. 50, 321-377 (1997) · Zbl 0881.35099 |
| [40] | Gesztesy, F., Grosse, H., Thaller, B.: A rigorious approach to relativistic corrections of bound state energies for spin-1/2 particles. Ann. Inst. Henri Poincaré Phys. Theor. 40, 159-174 (1984) · Zbl 0534.46053 |
| [41] | Gross, L.: The Cauchy problem for the coupled Maxwell and Dirac equations. Commun. Pure Appl. Math. 19, 1-15 (1966) · Zbl 0137.32401 · doi:10.1002/cpa.3160190102 |
| [42] | Huang, Z., Jin, S., Markowich, P.A., Sparber, C., Zheng, C.: A time-splitting spectral scheme for the Maxwell-Dirac system. J. Comput. Phys. 208, 761-789 (2005) · Zbl 1070.81040 · doi:10.1016/j.jcp.2005.02.026 |
| [43] | Hunziker, W.: On the nonrelativistic limit of the Dirac theory. Commun. Math. Phys. 40, 215-222 (1975) · doi:10.1007/BF01609998 |
| [44] | Jiang, S., Greengard, L., Bao, W.: Fast and accurate evaluation of nonlocal Coulomb and dipole – dipole interactions via the nonuniform FFT. SIAM J. Sci. Comput. 36, B777-B794 (2014) · Zbl 1307.65184 · doi:10.1137/130945582 |
| [45] | Jin, S., Markowich, P., Sparber, C.: Mathematical and numerical methods for semiclassical Schrödinger equations. Acta Numer. 20, 121-209 (2011) · Zbl 1233.65071 · doi:10.1017/S0962492911000031 |
| [46] | Lemou, M., Méhats, F., Zhao, X.: Uniformly accurate numerical schemes for the nonlinear Dirac equation in the nonrelativistic limit regime, arXiv:1605.02475 (2016) · Zbl 1457.35047 |
| [47] | McLachlan, R.I., Quispel, G.R.W.: Splitting methods. Acta Numer. 11, 341-434 (2002) · Zbl 1105.65341 · doi:10.1017/S0962492902000053 |
| [48] | Neto, A.H.C., Guinea, F., Peres, N.M.R., Novoselov, K.S., Geim, A.K.: The electronic properties of graphene. Rev. Mod. Phys. 81, 109-162 (2009) · doi:10.1103/RevModPhys.81.109 |
| [49] | Novoselov, K.S., Geim, A.K., Morozov, S.V., Jiang, D., Zhang, Y., Dubonos, S.V., Grigorieva, I.V., Firsov, A.A.: Electric field effect in atomically thin carbon films. Science 306, 666-669 (2004) · doi:10.1126/science.1102896 |
| [50] | Nraun, J.W., Su, Q., Grobe, R.: Numerical approach to solve the time-dependent Dirac equation. Phys. Rev. A 59, 604-612 (1999) · doi:10.1103/PhysRevA.59.604 |
| [51] | Ohlsson, T.: Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory. Cambridge University Press, Cambridge (2011) · Zbl 1314.81004 · doi:10.1017/CBO9781139032681 |
| [52] | Ring, P.: Relativistic mean field theory in finite nuclei. Prog. Part. Nucl. Phys. 37, 193-263 (1996) · doi:10.1016/0146-6410(96)00054-3 |
| [53] | Spohn, H.: Semiclassical limit of the Dirac equation and spin precession. Annal. Phys. 282, 420-431 (2000) · Zbl 1112.81320 · doi:10.1006/aphy.2000.6039 |
| [54] | Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 507-517 (1968) · Zbl 0184.38503 · doi:10.1137/0705041 |
| [55] | Suzuki, M.: Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Phys. Lett. A 146, 319-323 (1990) · doi:10.1016/0375-9601(90)90962-N |
| [56] | Suzuki, M.: General theory of fractal path integrals with applications to many body theories and statistical physics. J. Math. Phys. 32, 400-407 (1991) · Zbl 0735.47009 · doi:10.1063/1.529425 |
| [57] | Suzuki, M.: General decompositon theory of ordered exponentials. Proc. Japan Acad. 69, 161-166 (1993) · doi:10.2183/pjab.69.161 |
| [58] | Suzuki, M.: New scheme of hybrid exponential product formulas with applications to quantum Monte-Carlo Simulations. In: Springer Proceedings in Physics, vol. 80, pp. 169-174 (1995) |
| [59] | Thalhammer, M.: Convergence analysis of high-order time-splitting pseudo-spectral methods for nonlinear Schrödinger equations. SIAM J. Numer. Anal. 50, 3231-3258 (2012) · Zbl 1267.65116 · doi:10.1137/120866373 |
| [60] | Thaller, B.: The Dirac Equation. Springer, New York (1992) · Zbl 0881.47021 · doi:10.1007/978-3-662-02753-0 |
| [61] | Trotter, H.F.: On the product of semi-groups of operators. Proc. Am. Math. Soc. 10, 545-551 (1959) · Zbl 0099.10401 · doi:10.1090/S0002-9939-1959-0108732-6 |
| [62] | Wu, H., Huang, Z., Jin, S., Yin, D.: Gaussian beam methods for the Dirac equation in the semi-classical regime. Commun. Math. Sci. 10, 1301-1305 (2012) · Zbl 1281.65133 · doi:10.4310/CMS.2012.v10.n4.a14 |
| [63] | Xu, S., Belopolski, I., Alidoust, N., Neupane, M., Bian, G., Zhang, C., Sankar, R., Chang, G., Yuan, Z., Lee, C., Huang, S., Zheng, H., Ma, J., Sanchez, D.S., Wang, B., Bansil, A., Chou, F., Shibayev, P.P., Lin, H., Jia, S., Hasan, M.Z.: Discovery of a Weyl fermion semimetal and topological Fermi arcs. Science 349, 613-617 (2015) · doi:10.1126/science.aaa9297 |
| [64] | Yoshida, H.: Construction of higher order symplectic integrators. Phys. Lett. A 150, 262-268 (1990) · doi:10.1016/0375-9601(90)90092-3 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
