A Fast Method for Linear Waves Based on Geometrical Optics

We develop a fast method for solving the one-dimensional wave equation based on geometrical optics. From geometrical optics (e.g., Fourier integral operator theory or WKB approximation) it is known that high-frequency waves split into forward and backward propagating parts, each propagating with the wave speed, with amplitude that is slowly changing depending on the medium coefficients, under the assumption that the medium coefficients vary slowly compared to the wavelength. Based on this we construct a method of optimal, $O(N)$ complexity, with basically the following steps: 1. decouple the wavefield into an approximately forward and an approximately backward propagating part; 2. propagate each component explicitly along the characteristics over a time step that is small compared to the medium scale but can be large compared to the wavelength; 3. apply a correction to account for the errors in the explicit propagation; repeat steps 2 and 3 over the necessary amount of time steps; and 4. reconstruct the full field by adding forward and backward propagating components again. Due to step 3 the method accurately computes the full wavefield. A variant of the method was implemented and outperformed a standard order (4,4) finite difference method by a substantial factor. The general principle is applicable also in higher dimensions, but requires efficient implementations of Fourier integral operators which are still the subject of current research.

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