Orateur
Description
We highlight recent developments of a "Fourier continuation" method for the numerical analysis of parabolic/hyperbolic partial differential equations (PDEs) with complex boundary conditions. The framework relies on a discrete extension approach for the high-order trigonometric interpolation of non-periodic functions (mitigating the notorious Gibb’s “ringing effect"), enabling construction of FFT-speed solvers that can provide efficient resolution while faithfully preserving the dispersion/diffusion characteristics of the underlying continuous operators. We discuss treatment of variable-coefficient systems, curved geometries, Neumann-like (e.g., flux) boundary conditions, nonlinear/nonstationary coupling, and high-order PDEs (e.g., for immersed boundary lattice Boltzmann methods). The resulting solvers are accurate by means of relatively coarse discretizations; incur little-to-no numerical dispersion or diffusion errors; carry mild (linear) CFL constraints on time integration (less restrictive spectral radii than other pseudospectral methods); and can be efficiently parallelized.
The efficacy of these tools is demonstrated through collaborative problems in biofluids and geophysics.
