This talk discusses efforts to study a variety of time-dependent phenomena (in solids, fluids and their interactions) via the development of a new fast Fourier transform-based methodology for the numerical analysis of parabolic and hyperbolic partial differential equations (PDEs) with complex boundary conditions. Such a framework is based on a “continuation” approach for the high-order trigonometric interpolation of a non-periodic function (i.e., mitigating the notorious Gibb’s “ringing” effect), where the ultimate goal is to build high-performance PDE solvers on general domains that can provide stable and efficient resolution while faithfully preserving the dispersion characteristics of the underlying continuous problems. With an eye towards mutual validation of both simulation and experiment, the efficacy of these tools are demonstrated through some of the collaborative scientific problems that have inspired them, including those in materials science (ultrasonic non-destructive testing), cardiovascular medicine (pulsatile blood flow), and geophysics (seismogenic tsunamis).