High-order essentially nonsocillatory schemes for Hamilton-Jacobi equations
SIAM Journal on Numerical Analysis
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SIAM Journal on Numerical Analysis
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| Hi-index | 7.29 |
The aim of this work is the simulation of the acoustic propagation in a moving flow using the high-frequency approach. We linearize the Euler equations around a stationary state for which the resulting system of PDE cannot be in general reduced to a wave equation. We are however able to perform a high-frequency analysis of the acoustic perturbation, using the W.K.B. method, introducing a phase @f and an amplitude A. The phase @f is solution of a Hamilton-Jacobi equation that we solve by a numerical Eulerian method using a monotone scheme [S.J. Osher, C.W. Shu, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal, 28(4) (1991) 907-922] following Benamou et al. [A geometric optics method for high frequency electromagnetic fields computations near fold caustics Part I, J. Comput. Appl. Math. 156 (2003) 93-125]. Adopting the techniques of Lax and Rauch [Lectures on Geometric Optics, ] for hyperbolic systems, we compute the leading order term of the amplitude A. Our results are still valid in the neighborhood of a fold caustic.