Numerical integration methods for fast evaluation of the acoustic quasi-periodic Green’s function

Abstract

The purpose of this thesis is to investigate efficient calculation methods for integrals of the form I := Z ∞−∞e−ρv2F(v)dv, where ρ > 0 and F is a given analytic function, and to apply these calculation methods to particular integrals of this type that arise in applications in acoustics. It is well known that the simplest integration rule, the midpoint rule, is exponentially convergent as the step-size h → 0, if F is bounded and analytic in a strip surrounding the real axis, and that the contour integration arguments used to prove this lead to modifications to the midpoint rule that retain this exponential convergence in the case that F has pole singularities that may lie close to the real axis. In practice this midpoint rule has to be truncated. In the first part of this thesis we derive, by contour integration arguments, a new error estimate for a truncated version of the midpoint rule, modified with a correction factor to take into account simple poles of the integrand near the real axis. This estimate assumes that F is bounded on the real axis but not necessarily in a strip surrounding the real axis. In the second, larger part of the thesis, we consider the evaluation of the 2D acoustic quasiperiodic Green’s function, the solution to the problem of acoustic propagation from an infinite array of line sources in free space. We derive a new representation for this Green’s function, in terms of integrals of the above form, and apply the truncated modified midpoint rule to obtain concrete approximations. We compare our new approximations for this Green’s function with existing methods of evaluation, for the test examples selected in the review paper of Linton (J. Eng. Math. 33, 377- 401, 1998), and through more systematic testing over the full range of parameter values. We also use our general error estimate to derive a rigorous error estimate, as a function of the various parameters, for our new approximations to this Green’s function