Nonlinear Geometric Optics method based multi-scale numerical schemes for highly-oscillatory transport equations

We introduce a new numerical strategy to solve a class of oscillatory transport PDE models which is able to captureaccurately the solutions without numerically resolving the high frequency oscillations {\em in both space and time}.Such PDE models arise in semiclassical modeling of quantum dynamics with band-crossings, and otherhighly oscillatory waves. Our first main idea is to use the nonlinear geometric optics ansatz, which builds theoscillatory phase into an independent variable. We then choose suitable initial data, based on the Chapman-Enskog expansion, for the new model. For a scalar model, we prove that so constructed model will have certain smoothness, and consequently, for a first order approximation scheme we prove uniform error estimates independent of the (possibly small) wave length. The method is extended to systems arising from a semiclassical model for surface hopping, a non-adiabatic quantum dynamic phenomenon. Numerous numerical examples demonstrate that the method has the desired properties.

[1]  Moshe Gitterman,et al.  Semiclassical Approximation to Quantum Mechanics , 2012 .

[2]  G. MfiTIVIER.  Coherent and focusing multidimensional nonlinear geometric optics , 2018 .

[3]  Qian Niu,et al.  Berry phase effects on electronic properties , 2009, 0907.2021.

[4]  G. Hagedorn.  Raising and Lowering Operators for Semiclassical Wave Packets , 1998 .

[5]  Shi Jin.  ASYMPTOTIC PRESERVING (AP) SCHEMES FOR MULTISCALE KINETIC AND HYPERBOLIC EQUATIONS: A REVIEW , 2010 .

[6]  A. Majda,et al.  The validity of nonlinear geometric optics for weak solutions of conservation laws , 1985 .

[7]  March,et al.  Gaussian beams and the propagation of singularities , 2014 .

[8]  P. Markowich,et al.  On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime , 2002 .

[9]  Xiaowei Jia,et al.  A Uniformly Accurate Multiscale Time Integrator Pseudospectral Method for the Dirac Equation in the Nonrelativistic Limit Regime , 2015, SIAM J. Numer. Anal..

[10]  Stanley Osher,et al.  Numerical solution of the high frequency asymptotic expansion for the scalar wave equation , 1995 .

[11]  Nicolas Crouseilles,et al.  Asymptotic Preserving schemes for highly oscillatory Vlasov-Poisson equations , 2012, J. Comput. Phys..

[12]  Jeffrey Rauch,et al.  Hyperbolic Partial Differential Equations and Geometric Optics , 2012 .

[13]  Qin Li,et al.  A Multiband Semiclassical Model for Surface Hopping Quantum Dynamics , 2014, Multiscale Model. Simul..

[14]  M. Popov.  A new method of computation of wave fields using Gaussian beams , 1982 .

[15]  H. Kreiss.  Problems with different time scales for partial differential equations , 1980 .

[16]  Chunxiong Zheng,et al.  A time-splitting spectral scheme for the Maxwell-Dirac system , 2005, 1205.0368.

[17]  E Weinan,et al.  The heterogeneous multiscale method* , 2012, Acta Numerica.

[18]  N. R. Hill,et al.  Gaussian beam migration , 1990 .

[19]  Nicolas Crouseilles,et al.  Uniformly accurate numerical schemes for highly oscillatory Klein–Gordon and nonlinear Schrödinger equations , 2013, Numerische Mathematik.

[20]  Christof Sparber,et al.  Wigner functions versus WKB‐methods in multivalued geometrical optics , 2001 .

[21]  Claudia Negulescu,et al.  WKB-Based Schemes for the Oscillatory 1D Schrödinger Equation in the Semiclassical Limit , 2011, SIAM J. Numer. Anal..

[22]  Shi Jin,et al.  A hybrid Schrödinger/Gaussian beam solver for quantum barriers andsurface hopping , 2011 .

[23]  Erwan Faou,et al.  Computing Semiclassical Quantum Dynamics with Hagedorn Wavepackets , 2009, SIAM J. Sci. Comput..

[24]  Weizhu Bao,et al.  A Uniformly Accurate Multiscale Time Integrator Pseudospectral Method for the Klein-Gordon Equation in the Nonrelativistic Limit Regime , 2014, SIAM J. Numer. Anal..

[25]  Christof Sparber,et al.  Mathematical and computational methods for semiclassical Schrödinger equations* , 2011, Acta Numerica.

[26]  Mathieu Colin,et al.  Short Pulses Approximations in Dispersive Media , 2007, SIAM J. Math. Anal..

[27]  Stanley Osher,et al.  A level set method for the computation of multi-valued solutions to quasi-linear hyperbolic PDE's and Hamilton-Jacobi equations , 2003 .

[28]  S. Osher,et al.  A LEVEL SET METHOD FOR THE COMPUTATION OF MULTIVALUED SOLUTIONS TO QUASI-LINEAR HYPERBOLIC PDES AND HAMILTON-JACOBI EQUATIONS , 2003 .

[29]  Shi Jin,et al.  Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs Wigner , 2003 .

[30]  Eric J. Heller,et al.  Frozen Gaussians: A very simple semiclassical approximation , 1981 .

[31]  Shi Jin,et al.  Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations , 1999, SIAM J. Sci. Comput..

[32]  Jianfeng Lu,et al.  Convergence of frozen Gaussian approximation for high‐frequency wave propagation , 2010, 1012.5055.

[33]  Weizhu Bao,et al.  Uniformly accurate multiscale time integrators for highly oscillatory second order differential equations , 2012, 1212.4939.

[34]  Shi Jin,et al.  Gaussian beam methods for the Schrodinger equation in the semi-classical regime: Lagrangian and Eulerian formulations , 2008 .

[35]  O. Morandi,et al.  Wigner model for quantum transport in graphene , 2011, 1102.2416.

[36]  Shingyu Leung,et al.  Eulerian Gaussian Beams for High Frequency Wave Propagation , 2007 .

[37]  Andrew J. Majda,et al.  Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws , 1986 .

[38]  Shi Jin,et al.  Gaussian beam methods for the Schrödinger equation with discontinuous potentials , 2014, J. Comput. Appl. Math..

[39]  B. Engquist,et al.  Multi-phase computations in geometrical optics , 1996 .

[40]  C. Cercignani.  The Boltzmann equation and its applications , 1988 .

[41]  P. Degond.  Asymptotic-Preserving Schemes for Fluid Models of Plasmas , 2011, 1104.1869.

[42]  E Weinan,et al.  The Heterognous Multiscale Methods , 2003 .