A Fast Method for Solving the Helmholtz Equation Based on Wave Splitting

In this thesis, we propose and analyze a fast method for computing the solution of the Helmholtz equation in a bounded domain with a variable wave speed function. The method is based on wave splitting. The Helmholtz equation is first split into one–way wave equations which are then 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 GO with a computational cost independent of the frequency. Elsewhere, the equations are fully resolved with a Runge-Kutta method. We have been able to show rigorously in one dimension that the algorithm is convergent and that for fixed accuracy, the computational cost is just O(ω) for a p-th order Runge-Kutta method. This has been confirmed by numerical experiments.