A Multi-Frequency Helmholtz Solver Based on the WaveHoltz Algorithm

Abstract: We develop and analyze a new approach for simultaneously computing multiple solutions to the Helmholtz equation for different frequencies and different forcing functions. The new Multi-Frequency WaveHoltz (MFWH) algorithm is an extension of the original WaveHoltz method and both are based on time-filtering solutions to an associated wave equation. With MFWH, the different Helmholtz solutions are computed simultaneously by solving a single wave equation combined with multiple time filters. The MFWH algorithm defines a fixed-point iteration which can be accelerated with Krylov methods such as GMRES. The solution of the wave equation can be efficiently solved with either explicit time-stepping or implicit time-stepping using as few as five time-steps per period. When combined with an $O(N)$ solver for the implicit equations, such a multigrid, the scheme has an $O(N)$ solution cost when the frequencies are fixed and the number of grid points $N$ increases. High-order accurate approximations in space are used together with second-order accurate approximations in time. We show how to remove time discretization errors so that the MFWH solutions converge to the corresponding solutions to the discretized Helmholtz problems. Numerical results are given using second-order accurate and fourth-accurate discretizations to confirm the convergence theory.