A Numerical Comparison of Eigenfunction Matching and Singularity-Respecting Galerkin Approximation Methods for Linear Water Wave Scattering

The convergence of two numerical methods for solving linear water wave scattering problems, namely the eigenfunction matching method (EMM) and the singularity-respecting Galerkin approximation (SRGA), is examined. To do so, the methods are applied to two simple problems, namely scattering by a partially submerged vertical barrier and scattering in a parallel walled channel with a step change in width. These problems contain corner singularities in the velocity potential of order −1/2 and −1/3, respectively, which the SRGA accounts for but EMMs do not. The results presented to compare the methods show that SRGA solutions are consistently more accurate than EMM solutions for the same amount of computing time. The results also show that the EMM solution for the channel problem is more accurate than the EMM solution for the vertical barrier problem due to the weaker singularity. Nevertheless, the EMM for the barrier is shown to still converge beyond three decimal places if a sufficiently large matrix is used—slower computation may be a worthwhile trade-off in certain situations because the EMM is usually considered to be more straightforward to implement. Our results serve as a practical guide for researchers selecting between the numerical methods.