This contribution presents recent developments of a Fourier series-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 discrete "extension" 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, FFT-speed, dynamic PDE solvers on general (structured, curvilinear) geometries that can provide stable and efficient resolution while faithfully preserving the dispersion/diffusion characteristics of the underlying continuous operators.

In particular, the algorithms presented here extend the class of such methods (known as "Fourier continuation") to encompass variable-coefficient 3D systems, general (curved) 3D domains, 3D parallelization, Neumann-like (e.g., traction or convective flux) boundary conditions, nonlinear/nonstationary (ODE-governed) coupling (e.g., dynamic 0D-coupling), and applications to higher-order PDEs with multiple boundary conditions (e.g., Lagrangian mechanical formulations). The resulting solvers enjoy a number of desirable properties for scientific computation: accuracy by means of relatively coarse discretizations; little-to-no numerical dispersion or diffusion errors; mild (linear) CFL constraints on (explicit) time integration; and efficient parallelization for distributed-memory highperformance computing.

With an eye towards mutual validation of both simulation and experiment, the efficacy of the current state of these tools is demonstrated in the context of some of the collaborative scientific problems that have inspired them, including those in engineering (ultrasonic non-destructive testing and seismic wave propagation [1]), geophysics (seismogenic tsunamis [2]), and medicine (fluid-structure hemodynamics [3, 4], dye diffusion [5, 6]). Performance studies and comparisons to other solvers are additionally presented.