A fourth order operator marching method for the Helmholtz equation in a waveguide is developed in this paper. It is derived from a new fourth order exponential integrator for linear evolution equations. The method improves the second order accuracy associated with the widely used step-wise coupled mode method where the waveguide is approximated by segments that are uniform in the propagation di...

Boundary integral equations and Nyström discretization provide a powerful tool for the solution of Laplace and Helmholtz boundary value problems. However, often a weaklysingular kernel arises, in which case specialized quadratures that modify the matrix entries near the diagonal are needed to reach a high accuracy. We describe the construction of four different quadratures which handle logarith...

Abstract. The Ultra Weak Variational Formulation (UWVF) of the Helmholtz equation provides a variational framework suitable for discretization using plane wave solutions of the adjoint of the underlying equation. Currently convergence of the method is only proved on the boundary of the domain. However substantial computational evidence exists that shows that the method also converges throughout...

Capacitance matrix methods provide techniques for extending the use of fast Poisson solvers to arbitrary bounded regions. These techniques are further studied and developed with a focus on the three-dimensional case. A discrete analogue of classical potential theory is used as a guide in the design of rapidly convergent iterative methods. Algorithmic and programming aspects of the methods are a...

The main purpose of this article is to disseminate among a wide audience of physicists a known result, which is available since a couple of years to the cognoscenti of differential forms on manifolds; namely, that charge conservation implies the inhomogeneous Maxwell equations. This is the reciprocal statement of one which is very well known among physicists: charge conservation, written in the...

The performance of the boundary and finite element methods for the Helmholtz equation in two dimensions is investigated. To facilitate the comparison, the system of linear equations arising from the finite element formulation is reduced to a smaller system involving the boundary values of the unknown function and its normal derivative alone. The difference between the boundary and finite elemen...