Mathematics: Faculty Scholarship

Title

Error Estimates for Gaussian Beam Methods Applied to Symmetric Strictly Hyperbolic Systems

Article

7-10-2017

Keywords

Symmetric hyperbolic systems, Gaussian beams, Error estimates

Mathematics

Abstract

In this work we construct Gaussian beam approximations to solutions of linear symmetric hyperbolic systems with highly oscillatory initial data, including both strictly and non-strictly hyperbolic systems as long as they are diagonalizable. The evolution equations for each Gaussian beam component are derived. Under some regularity assumptions of the data we obtain an error estimate between the exact solution and the first order Gaussian beam superposition in terms of the high frequency parameter ε−1" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">ε−1. The main result is that the relative local error measured in energy norm in the beam approximation decays as ε12" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">ε12 independent of dimension and presence of caustics, for first order beams. This result is shown to be valid when the gradient of the initial phase is bounded away from zero. Applications to some non-strictly hyperbolic systems including both the acoustic equation and the system of Maxwell equations are discussed.

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