A wavefront-based Gaussian beam method for computing high frequency wave propagation problems

Abstract

a b s t r a c tWe present a novel wavefront method based on Gaussian beams for computing high frequency wave propagation problems. Unlike standard geometrical optics, Gaussian beams compute the correct solution of the wave field also at caustics. The method tracks a front of two canonical beams with two particular initial values for width and curvature. In a fast post-processing step, from the canonical solutions we recreate any other Gaussian beam with arbitrary initial data on the initial front. This provides a simple mechanism to include a variety of optimization processes, including error minimization and beam width minimization, for a posteriori selection of optimal beam initial parameters. The performance of the method is illustrated with two numerical examples.Published by Elsevier Ltd.

Sobolev and max norm error estimates for Gaussian beam superpositions

“…Numerical methods based on Gaussian beam type superpositions go back to the 1980's for the wave equation [31,15,2,17,39] and for the Schrödinger equation [6,7]. Since then a great many such methods have been developed for various applications [8,9,38,4,20,37,40,30,32]. Typically, the ODEs for the Taylor coefficients of the phase and amplitude are solved using numerical ODE methods like Runge-Kutta and the superposition integral (1.4) is approximated by the trapezoidal rule.…”

A Sparse Stochastic Collocation Technique for High-Frequency Wave Propagation with Uncertainty

“…where Z j (t, y) are disjoint open subsets of supp(A 0 ( · , y)). Based on this, let us assume that in the support of the QoI test function, ψ, there are no caustic points at t = T , and that the solution satisfies (26) and (27). The quantity of interest (7) can then be written…”

“…Indeed, suppose there is a stationary point, i.e., ∇φ i (T, x, y) = ∇φ j (T, x, y) for some x ∈ supp(ψ). Then, by (27), there are two geometrical optics rays starting at z i ∈ Z i (T, y) and z j ∈ Z j (T, y), respectively, such that But, by uniqueness of the solution to the Hamiltonian system (11), we must therefore have z i = z j . Hence, the two rays are the same and since the sets {Z j (t, y)} are disjoint, we get i = j, a contradiction.…”

“…Moreover, the phases, φ j , are related to the initial phase, Φ 0 , and to p(t, y; z) via the ray function, q(t, y; z), as follows. For each (t, x, y) ∈ U and each 1 ≤ j ≤ K, there is a z j ∈ Z j (t, y) such that x = q(t, y; z j ), φ j (t, x, y) = Φ 0 (z j , y), ∇φ j (t, x, y) = p(t, y; z j ), (27) where Z j (t, y) are disjoint open subsets of supp(A 0 ( • , y)).…”

“…The stochastic collocation method has been widely used in forward propagation of uncertainty in many PDE models, see, e.g., [40,2,31,30,4,24,8,12,1]. The Gaussian beam summation method has also been successfully applied to deterministic high-frequency wave propagation problems, see, e.g., [33,35,25,26,15,18,27]. However, combining the two methods is not straightforward since the convergence rate of the resulting method strongly depends on the stochastic regularity of the QoI, which in general depends on ε and in principle may be as bad as the regularity of u ε .…”