# zbMATH — the first resource for mathematics

Exact splitting methods for kinetic and Schrödinger equations. (English) Zbl 1459.35341
The authors discuss some PDE involving a differential operator $$p^w$$ whose symbol (in the Weyl quantization) is a polynomial function of degree 2 or less. It is said that the operator $$p^w$$ can be computed by exact splitting if it can be factored into operators having a special basic form (see Definition 1 in the paper).
Having an exact splitting in hand it is possible to discretize the PDE in question, by solving via pseudo-spectral methods or pointwise multiplications, and the discretized solution will converge to an actual solution of the PDE.
The authors discuss transport equations, Focker-Planck equations, magnetic Schrödinger equations and rotating Gross-Pitaevskii equations. Each of these equations involve an operator with a quadratic symbol, for which the authors prove the exact splitting along with numerical results on the solutions.
##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q49 Transport equations 35Q84 Fokker-Planck equations 82C40 Kinetic theory of gases in time-dependent statistical mechanics 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs
GPELab
Full Text:
##### References:
 [1] Alphonse, P., Bernier, J.: Polar decomposition of semigroups generated by non-selfadjoint quadratic differential operators and regularizing effects. arXiv:1909.03662 (2019) [2] Ameres, J.: Splitting methods for Fourier spectral discretizations of the strongly magnetized Vlasov-Poisson and the Vlasov-Maxwell system. arXiv preprint arXiv:1907.05319 (2019) [3] Antoine, X.; Bao, W.; Besse, C., Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Commun., 184, 12, 2621-2633 (2013) · Zbl 1344.35130 [4] Antoine, X.; Duboscq, R., GPELab, a Matlab toolbox to solve Gross-Pitaevskii equations I: computation of stationary solutions, Comput. Phys. Commun., 185, 11, 2969-2991 (2014) · Zbl 1348.35003 [5] Antoine, X.; Duboscq, R., Gpelab, a matlab toolbox to solve Gross-Pitaevskii equations II: dynamics and stochastic simulations, Comput. Phys. Commun., 193, 95-117 (2015) · Zbl 1344.82004 [6] Bader, P., Fourier-splitting methods for the dynamics of rotating Bose-Einstein condensates, J. Comput. Appl. Math., 336, 267-280 (2018) · Zbl 1382.65462 [7] Bader, P.; Blanes, S., Fourier methods for the perturbed harmonic oscillator in linear and nonlinear Schrödinger equations, Phys. Rev. E, 83, 4, 046711 (2011) [8] Bader, P., Blanes, S., Casas, F.: Efficient time integration methods for Gross-Pitaevskii equations with rotation term. preprint arXiv:1910.12097 (2019) · Zbl 1434.35175 [9] Bao, W.; Du, Q.; Zhang, Y., Dynamics of rotating Bose-Einstein condensates and its efficient and accurate numerical computation, SIAM J. Appl. Math., 66, 3, 758-786 (2006) · Zbl 1141.35052 [10] Bao, W.; Marahrens, D.; Tang, Q.; Zhang, Y., A simple and efficient numerical method for computing the dynamics of rotating Bose-Einstein condensates via rotating Lagrangian coordinates, SIAM J. Sci. Comput., 35, 6, A2671-A2695 (2013) · Zbl 1286.35213 [11] Bao, W.; Wang, H., An efficient and spectrally accurate numerical method for computing dynamics of rotating Bose-Einstein condensates, J. Comput. Phys., 217, 2, 612-626 (2006) · Zbl 1160.82343 [12] Bernier, J.: Exact splitting methods for semigroups generated by inhomogeneous quadratic differential operators. Preprint, arXiv:1912.13219 (2019) [13] Bernier, J.; Casas, F.; Crouseilles, N., Splitting methods for rotations: application to Vlasov equations, SIAM J. Sci. Comput., 42, 2, 1 (2020) · Zbl 1432.65133 [14] Besse, C.; Descombes, S.; Dujardin, G.; Lacroix-Violet, I., Energy preserving methods for nonlinear Schrödinger equations, IMA J. Numer. Anal., 1, drz067 (2020) [15] Besse, C.; Dujardin, G.; Lacroix-Violet, I., High order exponential integrators for nonlinear Schrödinger equations with application to rotating Bose-Einstein condensates, SIAM J. Numer. Anal., 55, 3, 1387-1411 (2017) · Zbl 1371.35235 [16] Besse, N.; Mehrenberger, M., Convergence of classes of high-order semi-Lagrangian schemes for the Vlasov-Poisson system, Math. Comput., 77, 261, 93-123 (2008) · Zbl 1131.65080 [17] 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 [18] Chen, B.; Kaufman, A., 3D volume rotation using shear transformations, Graph. Models, 62, 4, 308-322 (2000) [19] Coulaud, O.; Sonnendrücker, E.; Dillon, E.; Bertrand, P., J. Plasma Phys., 61, 435-448 (1999) [20] Dujardin, G., Hérau, F., Lafitte, P.: Coercivity, hypocoercivity, exponential time decay and simulations for discrete Fokker-Planck equations. arXiv preprint arXiv:1802.02173 (2018) · Zbl 1435.35385 [21] Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics (2006) · Zbl 1094.65125 [22] Hérau, F.; Sjöstrand, J.; Hitrik, M., Tunnel effect for the Kramers-Fokker-Planck type operators: return to equilibrium and applications, Int. Math. Res. Not., 57, 48 (2008) · Zbl 1151.35012 [23] Hérau, F.; Sjöstrand, J.; Hitrik, M., Tunnel effect for the Kramers-Fokker-Planck type operators, Ann. Henri Poincaré, 9, 209-274 (2008) · Zbl 1141.82011 [24] Hérau, F.; Thomann, L., On global existence and trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with exterior confining potential, J. Funct. Anal., 271, 1301-1340 (2016) · Zbl 1347.35221 [25] Hochbruck, M.; Ostermann, A., Exponential integrators, Acta Numerica, 19, 209-286 (2010) · Zbl 1242.65109 [26] Hörmander, L., Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z., 219, 413-449 (1995) · Zbl 0829.35150 [27] Hörmander, L.: The analysis of linear partial differential operators. III, classics in mathematics. Pseudo-differential operators. Springer, Berlin (2007). doi:10.1007/978-3-540-49938-1 · Zbl 1115.35005 [28] Jin, S.; Zhou, Z., A semi-Lagrangian time splitting method for the Schrödinger equation with vector potentials, Commun. Inf. Syst., 13, 3, 247-289 (2013) · Zbl 1305.35122 [29] Li, Y.; He, Y.; Sun, Y., Solving the Vlasov-Maxwell equations using Hamiltonian splitting, J. Comput. Phys., 396, 381-399 (2019) [30] McLachlan, RI; Quispel, GR, Splitting methods, Acta Numerica, 11, 341-434 (2002) · Zbl 1105.65341 [31] Marsden, JE; Ratiu, TS, Introduction to mechanics and symmetry: a basic exposition of classical mechanical systems (2013), Berlin: Springer, Berlin · Zbl 0933.70003 [32] Raymond, N.: Bound States of the Magnetic Schrödinger Operator. EMS Tracts Math. (2017) · Zbl 1370.35004 [33] Nicola, F.; Rodino, L., Global Pseudo-Differential Calculus on Euclidean Spaces (2010), Basel: Birkhäuser, Basel · Zbl 1257.47002 [34] Welling, JS; Eddy, WF; Young, TK, Rotation of 3D volumes by Fourier-interpolated shears, Graph. Models, 68, 4, 356-370 (2006) · Zbl 1103.68916 [35] Zeng, R.; Zhang, Y., Efficiently computing vortex lattices in rapid rotating Bose-Einstein condensates, Comput. Phys. Commun., 180, 6, 854-860 (2009) · Zbl 1198.82007
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.