We present and compare two different techniques to obtain the multi-phase solutions for the Schrödinger equation in the semiclassical limit. The first is Whitham's averaging method, which gives the modulation equations governing the evolution of multi-phase solutions. The second is the Wigner transform, a convenient tool to derive the semiclassical limit equation in the phase space (the Vlasov equation) for the linear Schrödinger equation. Motivated by the linear superposition principle, we derive and prove the multi-phase ansatz for the Wigner function by the stationary phase method, and then use the ansatz to close the moment equations of the Vlasov equation and obtain the multi-phase equations in the physical space. We show that the multi-phase equations so derived agree with those derived by Whitham's averaging method, which can be proved using different arguments. Generic way of obtaining and computing the multi-phase equations by the Wigner function is given, and kinetic schemes are introduced to solve the multi-phase equations. The numerical schemes are purely Eulerian and only operate in the physical space. Several numerical examples are given to explore the validity of this approach. Similar studies are conducted for the linearized Korteweg-de Vries equation and the linear wave equation.

We develop a level set method for the computation of multivalued solutions to quasi-linear hyperbolic partial di#erential equations and Hamilton-Jacobi equations in any number of space dimensions. We use the classic idea of Courant and Hilbert to define the solution of the quasi-linear hyperbolic PDEs or the gradient of the solution to the Hamilton-Jacobi equations as zero level sets of level set functions. Then the evolution equations for the level set functions satisfy linear Liouville equations defined in the "phase" space, unfolding the singularities and preventing the numerical capturing of the viscosity solution. This provides a computational framework for the computations of multivalued geometric solutions to general quasilinear PDEs. By using the local level set method the cost of each time update for this method is O(N log N) for a d dimensional problem, where N is the number of grid points in each dimension.

We present a computational approach for the WKB approximation of the wave function of an electron moving in a periodic one-dimensional crystal lattice. We derive a non-strictly hyperbolic system for the phase and the intensity where the flux functions originate from the Bloch spectrum of the Schrodinger operator. Relying on the framework of the multibranch entropy solutions introduced by Brenier and Corrias, we compute e#ciently multiphase solutions using well adapted and simple numerical schemes. In this first part we present computational results for vanishing exterior potentials which demonstrate the e#ectiveness of the proposed method.

We apply the level set method to compute the three dimensional multivalued geometrical optics term in the paraxial formulation. The paraxial formulation is obtained from the 3-D stationary eikonal equation by using one of the spatial directions as the artificial evolution direction. The advection velocity field used to move level sets is obtained by the method of characteristics; therefore the motion of level sets is defined in a phase space. The multivalued traveltime and amplitude-related quantity are obtained from solving advection equations with source terms. We derive an amplitude formula in the reduced phase space which is very convenient to use in the level set framework. By using a semi-Lagrangian method in the paraxial formulation, the method has O(N²) rather than O(N^4) memory storage requirement in the five dimensional phase space, where N is the number of mesh points along one direction. Although the computational complexity is still O(MN^4), where M is the number of steps in the ODE solver for the semi-Lagrangian scheme, this disadvantage is largely overcome by the fact that up to O(n²) multiple sources can be treated simultaneously. Three dimensional numerical examples demonstrate the efficiency and accuracy of the method.

We develop a numerical scheme for the wave front computation of com-plete transmissions and reflections in geometrical optics. Such a problem can be formulated by a reduced Liouville equation with a discontinuous local wave speed or index of refraction, arising in the high frequency limit of linear waves through inhomogeneous media. The key idea is to incorporate Snell’s Law of Refraction into the numerical flux for the reduced Liouville equation. This scheme allows a hyperbolic CFL condition, under which positivity, and sta-bilities in both l ∞ and l 1 norms, are established. Numerical experiments are carried out to demonstrate the validity and accuracy of this new scheme. 1

The solution to the Schrödinger equation is highly oscillatory when the rescaled Planck constant ε is small in the semiclassical regime. A direct numerical simulation requires the mesh size to be O(ε). The Gaussian beam method is an efficient way to solve the high frequency wave equations asymptotically, outperforming the geometric optics method in that the Gaussian beam method is accurate even at caustics. In this paper, we solve the Schrödinger equation using both the Lagrangian and Eulerian formulations of the Gaussian beam methods. A new Eulerian Gaussian beam method is developed using the level set method based only on solving the (complex-valued) homogeneous Liouville equations. A major contribution here is that we are able to construct the Hessian matrices of the beams by using the level set function’s first derivatives. This greatly reduces the computational cost in computing the Hessian of the phase function in the Eulerian framework, yielding an Eulerian Gaussian beam method with computational complexity comparable to that of the geometric optics but with a much better accuracy around caustics. We verify through several numerical experiments that our Gaussian beam solutions are good approximations to Schrödinger solutions even at caustics. We also numerically study the optimal relation between the number of beams and the rescaled Planck constant ε in the Gaussian beam summation.

We apply the level-set methodology to compute multivalued solutions of the paraxial eikonal equation in both isotropic and anisotropic metrics. This paraxial equation is obtained from 2-D stationary eikonal equations by using one of the spatial directions as the arti cial evolution direction. The advection velocity eld used to move level sets is obtained by the method of characteristics; therefore the motion of level sets is de ned in a phase space, and the zero level set yields the location of bicharacteristic strips in the reduced phase space. The multivalued traveltime is obtained from solving another advection equation with a source term. The complexity of the algorithm is O(N LogN) in the worst case and O(N as the typical Lagrangian ray-tracing, where N is the number of the sampling points along one of the spatial directions. Numerical experiments including the well-known Marmousi synthetic model illustrate the accuracy and the eciency of the Eulerian method.

We propose a local level set method for constructing the geometrical optics term in the paraxial formulation for the high frequency asymptotics of 2-D acoustic wave equations. The geometrical

Development of level set methods for computing the semiclassical limit of Schrödinger equations with potentials by

Hamiltonian-preserving schemes for the Liouville equation of geometrical optics with discontinuous local wave speeds ∗