A Novel Adaptive Spectral Method for Fluid Flow Simulations in Non-periodic Domains

We present a novel spectral method for computing fluid flow in a domain with non-periodic boundaries. Unlike the spectral element or the discontinuous Galerkin method, which rely on polynomial expansions within each element, our method relies on a global expansion, wherein we decompose the flow variables into periodic and aperiodic parts. While the periodic part is represented through the standard Fourier basis (normal modes), the aperiodic part is represented through an adaptive basis of complex exponentials involving non-normal modes. To compute the derivatives, the periodic part utilizes standard Fast Fourier Transform (FFT) and the aperiodic part utilizes a low-rank approximation of arbitrary (tunable) accuracy, typically involving O(1) modes. The non-normal modes that span the basis for the aperiodic part are computed adaptively based on the jumps in the variable and its derivatives across the domain boundaries. Notably, the present method does not require extension of the computational domain unlike existing methods such as the Fourier extension method. We demonstrate the resolving efficacy of the method with suitable examples.