Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs Wigner

Bao, 2002, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175, 487, 10.1006/jcph.2001.6956 Benamou, 1996, Big ray tracing: multivalued travel time field computation using viscosity solution of the eikonal equation, J. Comput. Phys., 128, 463, 10.1006/jcph.1996.0224 Benamou, 1999, Direct solution of multivalued phase space solutions for Hamilton–Jacobi equation, Commun. Pure Appl. Math., 52, 1443, 10.1002/(SICI)1097-0312(199911)52:11<1443::AID-CPA3>3.0.CO;2-Y C. Bender, S. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978. F. Bouchut, Shi Jin, X.T. Li, Numerical approximations of pressureless gas and isothermal gas dynamics, SIAM J. Num. Anal. 41 (2003) 135–158. Brenier, 1998, A kinetic formulation for multibranch entropy solutions of scalar conservation laws, Ann. Inst. Henry Poincaré, 15, 169, 10.1016/S0294-1449(97)89298-0 Y. Brenier, L. Corrias, Capturing multi-valued solutions, preprint. Brenier, 1998, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35, 2317, 10.1137/S0036142997317353 J.J. Duistermaat, Fourier Integral Operators, Birkhäuser, Basel, 1995. W. E, Y.G. Rykov, Y.G. Sinai, Generalized variational principle, global weak solutions and behavior with random initial data, Commun. Math. Phys. 177 (1996) 349–380. Engquist, 1995, Numerical solution of the high frequency asymptotic expansion for the scalar wave equation, J. Comput. Phys., 120, 145, 10.1006/jcph.1995.1154 Engquist, 1996, Multi-phase computations in geometrical optics, J. Comput. Appl. Math., 74, 175, 10.1016/0377-0427(96)00023-4 B. Engquist, O. Runborg, A.-K. Tornberg, High frequency wave propagation by the segment projection methods, UCAL CAM Report 01-13, 2001. L. Evans, Partial Differential Equations, AMS Press, 1998. Flaschka, 1980, Multi-phase averaging and the inverse spectral solution of the Korteweg–de Vries equation, Commun. Pure Appl. Math., 33, 739, 10.1002/cpa.3160330605 M.G. Forest, J.E. Lee, Geometry and modulation theory for the periodic nonlinear Schrödinger equation, Oscillation Theory, Computation, and Methods of Compensated Compactness, Minneapolis, Minnesota, 1985, pp. 35–69, IMA Volume of Mathematics with Applications, vol. 2, Springer, New York, 1986. Gasser, 1997, Quantum hydrodynamics, Wigner transforms and the classical limits, Asymptotic Anal., 14, 97, 10.3233/ASY-1997-14201 Gerard, 1997, Homogenization limits and Wigner transforms, Commun. Pure Appl. Math., 50, 323, 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C Gosse, 2002, Using K-branch entropy solutions for multivalued geometric optics computations, J. Comput. Phys., 180, 155, 10.1006/jcph.2002.7085 L. Gosse, Shi Jin, X.T. Li, Two Moment Systems for computing multi-phase semiclassical limits of the Schrödinger equation, Math. Models Methods Appl. Sci., submitted. T. Grava, From the solution of the Tsarev system to the solution of the Whitham equations, Math. Phys. Anal. Geom. 4 (2001) 65–96. Shan Jin, C.D. Levermore, D.W. McLaughlin, The behavior of solutions of the NLS equations in the semiclassical limit, Singular Limits of Dispersive Waves, Lyon, Nato Adv. Sci. Inst. Ser. B Phys. 320, Plenum, New York, 1991, pp. 235–255. Jin, 1999, The semiclassical limit of the defocusing NLS hierarchy, Commun. Pure Appl. Math., 52, 613, 10.1002/(SICI)1097-0312(199905)52:5<613::AID-CPA2>3.0.CO;2-L L.D. Landau, E.M. Lifschitz, Quantum Mechanics: Non-relativistic Theory, Pergamon Press, New York, 1977. Lax, 1983, The small dispersion limit of the Korteweg–de Vries equation. III, Commun. Pure Appl. Math., 36, 253, 10.1002/cpa.3160360302 Lax, 1983, The small dispersion limit of the Korteweg–de Vries equation. III, Commun. Pure Appl. Math., 36, 571, 10.1002/cpa.3160360503 Lax, 1983, The small dispersion limit of the Korteweg–de Vries equation. III, Commun. Pure Appl. Math., 36, 809, 10.1002/cpa.3160360606 X.T. Li, Ph.D. Thesis, University of Wisconsin-Madison, 2002. Lions, 1993, Sur les measures de Wigner, Revista. Mat. Iberoamericana, 9, 553, 10.4171/RMI/143 Lions, 1995, Convergence of MUSCL and filtered schemes for scalar conservation laws and Hamilton–Jacobi equations, Numer. Math., 69, 441, 10.1007/s002110050102 Markowich, 1994, A Wigner function approach to semiclassical limits: electrons in a periodic potential, J. Math. Phys., 35, 1066, 10.1063/1.530629 Markowich, 1999, Numerical approximation of quadratic observables of Schrödinger-type equations in the semiclassical limit, Numer. Math., 81, 595, 10.1007/s002110050406 P.A. Markowich, P. Pietra, C. Pohl, H.P. Stimming, A Wigner-Measure analysis of the Dufort-Frankel scheme for the Schrödinger equation, SIAM J. Num. Anal. 40 (2002) 1281–1310. V.P. Maslov, Semiclassical Approximation in Quantum Mechanics, Reidel, Dordrecht, 1981. S. Osher, L.T. Cheng, M. Kang, H. Shim, Y.H. Tsai, Geometric optics in a phase space based level set and Eulerian framework, J. Comp. Phys. 79 (2002) 622–648. G. Papanicolaou, L. Ryzhik, Waves and Transport, IAS/Park City Mathematics Series, vol. 5, 1999. Runborg, 2000, Some new results in multi-phase geometrical optics, Math. Model Numer. Anal., 34, 1203, 10.1051/m2an:2000124 Sheng, 1999, The Riemann problem for the transportation equations in gas dynamics, Mem. Am. Math. Soc., 137, 194 C. Sparber, A Wigner function approach to (multivalued) geometrical optics, M.S. Thesis, Wien, February 2001. C. Sparber, M. Markowich, N. Mauser, Multivalued geometrical optics: Wigner vs. WKB, Asymptotic Analysis 33 (2003) 153–187. Venakides, 1985, The generation of modulated wavetrains in the solution of the Korteweg–de Vries equation, Commun. Pure Appl. Math., 38, 883, 10.1002/cpa.3160380616 Wang, 1997, Uniqueness of generalized solution for the Cauchy problem of transportation equations, Acta Math. Sci. English Ed., 17, 341, 10.1016/S0252-9602(17)30852-4 Whitham, 1965, Non-linear dispersive waves, Proc. Roy. Soc., Ser. A, 283, 238, 10.1098/rspa.1965.0019 G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. Wigner, 1932, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40, 749, 10.1103/PhysRev.40.749