Euan A. Spence

Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering

In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles. These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions. They have the potential to solve scattering problems accurately in a computation time that is (almost) independent of frequency and this has been realized for many model problems. The design and analysis of this class of methods requires new results on the analysis and numerical analysis of highly oscillatory boundary integral operators and on the high-frequency asymptotics of scattering problems. The implementation requires the development of appropriate quadrature rules for highly oscillatory integrals. This article contains a historical account of the development of this currently very active field, a detailed account of recent progress and, in addition, a number of original research results on the design, analysis and implementation of these methods.

Synthesis, as Opposed to Separation, of Variables

Every applied mathematician has used separation of variables. For a given boundary value problem (BVP) in two dimensions, the starting point of this powerful method is the separation of the given PDE into two ODEs. If the spectral analysis of either of these ODEs yields an appropriate transform pair, i.e., a transform consistent with the given boundary conditions, then the given BVP can be reduced to a BVP for an ODE. For simple BVPs it is straightforward to choose an appropriate transform and hence the spectral analysis can be avoided. In spite of its enormous applicability, this method has certain limitations. In particular, it requires the given domain, PDE, and boundary conditions to be separable, and also may not be applicable if the BVP is non-self-adjoint. Furthermore, it expresses the solution as either an integral or a series, neither of which are uniformly convergent on the boundary of the domain (for nonvanishing boundary conditions), which renders such expressions unsuitable for numerical computations. This paper describes a recently introduced transform method that can be applied to certain nonseparable and non-self-adjoint problems. Furthermore, this method expresses the solution as an integral in the complex plane that is uniformly convergent on the boundary of the domain. The starting point of the method is to write the PDE as a one-parameter family of equations formulated in a divergence form, and this allows one to consider the variables together. In this sense, the method is based on the “synthesis” as opposed to the “separation” of variables. The new method has already been applied to a plethora of BVPs and furthermore has led to the development of certain novel numerical techniques. However, a large number of related analytical and numerical questions remain open. This paper illustrates the method by applying it to two particular non-self-adjoint BVPs: one for the linearized KdV equation formulated on the half-line, and the other for the Helmholtz equation in the exterior of the disc (the latter is non-self-adjoint due to the radiation condition). The former problem played a crucial role in the development of the new method, whereas the latter problem was instrumental in the full development of the classical transform method. Although the new method can now be presented using only classical techniques, it actually originated in the theory of certain nonlinear PDEs called integrable, whose crucial feature is the existence of a Lax pair formulation. It is shown here that Lax pairs provide the generalization of the divergence formulation from a separable linear to an integrable nonlinear PDE.

Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed?

There has been much recent research on preconditioning discretisations of the Helmholtz operator

A new transform method II: the global relation and boundary-value problems in polar coordinates

\Delta + k^2

Is the Helmholtz Equation Really Sign-Indefinite? ∗

Δ+k2 (subject to suitable boundary conditions) using a discrete version of the so-called “shifted Laplacian”

Recent Results on Domain Decomposition Preconditioning for the High-Frequency Helmholtz Equation Using Absorption

\Delta + (k^2+ \mathrm{i}\varepsilon )