Analysis of a fast method for solving the high frequency Helmholtz equation in one dimension (Q639962)

Based on wave splitting, the authors introduce a fast method for computing the solution of the high frequency Helmholtz equation in a bounded one-dimensional domain with a variable wave speed function. The Helmholtz equation is split into one-way wave equations with source functions which are solved iteratively for a given tolerance. The source functions depend on the wave speed function and on the solutions of the one-way wave equations from the previous iteration. The solution of the Helmholtz equation is then approximated by the sum of the one-way solutions at every iteration. To improve the computational cost, the source functions are thresholded and in the domain where they are equal to zero, the one-way wave equations are solved with geometrical optics with a computational cost independent of the frequency. Convergence of the algorithm is proved in one dimension. Numerical experiments indicate that the growth rate of the computational cost is much slower than a direct method.