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.

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.

Wavefront construction in geometrical optics has long faced the twin difficulties of dealing with multivalued forms and resolution of wavefront surfaces. A recent change in viewpoint, however, has demonstrated that working in phase space on bicharacteristic strips using eulerian methods can bypass both difficulties. The success of the level set method in science and engineering makes it a suitable choice for such an eulerian method. Unfortunately, in three-dimensional space, the setting of interest for most practical applications, the advantages of this method are largely offset by a new problem: the high dimension of phase space. In this work, we present new types of level set algorithms that remove this obstacle and demonstrate their abilities to accurately construct wavefronts under high resolution. These results propel the level set method forward significantly as a competitive approach in geometrical optics under realistic conditions. 1

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Solving the Einstein constraint equations on multi-block triangulations using finite element methods