Browsing by Author "Aldahham, Ebraheem K.H. A. H."

Filter results by typing the first few letters
Now showing 1 - 1 of 1
  • Thumbnail Image
    Item
    Non-standard numerical methods for time-dependent PDE in unbounded domains
    (Heriot-Watt University, 2023-12) Aldahham, Ebraheem K.H. A. H.; Banjai, Professor Lehel
    We describe a series of numerical methods for the solution of acoustic exterior scattering problems based on the time-domain boundary integral representation of the solution. As the spatial discretization of the resulting time-domain boundary integral equation we use either the method of fundamental solutions (MFS) or the Galerkin boundary element method (BEM). In time we apply either a standard convolution quadrature (CQ) based on an A-stable linear multistep method or a modified CQ scheme. It is well-known that the standard low-order CQ schemes for hyperbolic problems suffer from strong dissipation and dispersion properties. The modified scheme is designed to avoid these properties. We give a careful description of the modified scheme and its implementation with differences due to different spatial discretizations highlighted. Numerous numerical experiments illustrate the effectiveness of the modified scheme and dramatic improvement with errors up to two orders of magnitude smaller in comparison with the standard scheme. Further, we combine the convolution spline method in time and MFS in space, revealing a higher convergence rate. However, this approach presents instability if the MFS source point radius is not sufficiently close to the boundary, highlighting the stability of the modified CQ due to the use of FFT in the implementation of the scheme. This observation leads us into an investigation of the underlying cause of this stability, which is highlighted in this thesis. In the final chapter, we address the challenges of high-frequency scattering by employing the physical optics approximation to reformulate the problem, resulting in an unknown function that oscillates slowly in space. This reformulation has been shown to be robust against frequency variations and is specifically applied within the methodological constraints of convex domains. Through this study, we gain a better understanding of the problem of acoustic scattering and lay the foundation for future research.