An augmented streamline upwind/Petrov-Galerkin method for the time-spectral convection-diffusion equation

Discretizing a solution in the spectral rather than time domain presents a significant advantage in solving transport problems encountered in fields like cardiorespiratory modeling, where the flow varies smoothly and periodically in time. To solve the system expressed in the frequency domain, one may rely on the classical time domain upwind techniques, such as the streamline upwind/Petrov-Galerkin (SUPG). While these classical methods successfully remove spurious oscillations in the solution in convection dominated flows, their accuracy deteriorates in a time-spectral setting as the element Womersley number approaches one. To overcome this limitation, this study introduces a new stabilized method, which we call augmented SUPG (ASU). The ASU is a consistent weighted residual method with two complex-valued stabilization parameters that act independently on the source and convective trial functions. Through a series of test cases, the superior accuracy of the ASU in comparison to four classical methods is shown across a wide range of flow conditions.