This thesis considers algorithmic and computational acceleration of numerical wave modelling at high frequencies. Numerical propagation of linear waves at high frequencies poses a significant challenge to modern simulation techniques. Despite the fact that potential practical benefits led a great deal of attention to this problem, current research has yet to provide a general and performant method to solve it. I consider finite element as a possible solution because it can handle geometric complex- ity of heterogeneous domains, but unfortunately it suffers from the “pollution” effect which imposes a prohobitively large memory requirement to handle high frequencies. One recent step towards enabling the finite element method to solve high frequency wave propagation in the frequency domain involves using a plane-wave basis rather than the standard polynomial basis. This allows highly compressed representations of scattering waves but otherwise appeared to limit users to nearly-homogeneous prob- lems. This thesis explores the use of plane-waves in a discontinuous Galerkin method (PWDG) for highly heterogeneous problems possibly containing a point source. The low-memory nature of PWDG and the fact that its expressions can be computed in an entirely symbolic manner without quadratures furthermore permits an efficient graphics processing unit (GPU) implementation such that problems with very high frequencies can be solved on a single workstation. This thesis includes computational results demonstrating results for frequencies in excess of 100 hertz on the Marmousi model, solved using only a single GPU.