A WKB based preconditioner for the 1D Helmholtz wave equation

Frequency-domain full-wave solutions to the cold-plasma problem have become ubiquitous in the study of radio frequency power in fusion plasmas. However, recent efforts at extreme levels of geometric fidelity have revealed fundamental limits in the problem size that can be solved by typical sparse direct solver based methods. These limits are of particular importance in the 3D study of RF launchers, where the number of degrees of freedom required can exceed 100 million. In such cases, it would be advantageous to solve the system via iterative means, but due to the large null space of the curl-curl operator, the convergence properties of algorithms like GMRES are poor. Here we present a physics-based preconditioner in the form of a WKB solution and demonstrate the iterative solution to the frequency-domain Helmholtz problem in 1D for several cases ranging from satisfying the WKB approximation to strongly violating it.