The numerical solution of highly oscillatory wave-propagation and scattering problems presents a variety of significant challenges: these problems require high discretization densities and often give rise to poorly conditioned numerics; realistic engineering configurations, further, usually require consideration of geometries of great complexity and large extent. In this talk we will consider a number of methodologies that were introduced recently to address these difficulties. We will thus discuss algorithms that can solve, with high-order accuracy, problems of scattering for complex three-dimensional geometries---including, possibly, singular elements such as wires, corners, edges and open screens. In particular, we will describe solutions achieved for two realistic three-dimensional problems of very high frequency---surface scattering and atmospheric GPS propagation---which previous three-dimensional solvers could not address adequately.
For added efficiency, these solvers, which are based on integral equations, high-order integration and fast Fourier transforms, can be used in conjunction with new regularized combined field equations---which require much smaller numbers of iterations in a iterative linear algebra solver than combined field equations available previously. We will also describe a new class of high-order surface representation methods which, starting from point clouds or CAD data, can produce high-order-accurate surface parametrizations of complex engineering surfaces, as required by high-order solvers. Time permitting, applications of these methodologies to solution of time-domain problems and fast evaluation of fully-nonlocal and convergent computational boundary conditions for time-domain problems will be mentioned. In all cases these algorithms exhibit high-order convergence, they run on low memories and reduced operation counts, and they can produce solutions with a high degree of accuracy.