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Space-time spectral methods for the Navier-Stokes equations
The Navier-Stokes equations describe the motion of viscous fluids, which are extensively used in engineering and physics to model fluid flow phenomena, such as the aerodynamics of aircraft, weather prediction, oil and gas pipeline design, and optimization of industrial processes involving fluid dynamics. They consist of a set of nonlinear partial differential equations that govern the conservation of mass, momentum, and energy in a fluid system.
In this talk, we present two space-time spectral methods for solving the unsteady Navier-Stokes problems. The first method employs a $P_N-P_{N-2}$ scheme, and the second method utilizes a staggered grid collocation scheme with quadrature nodes such as the Jacobi Gauss, Jacobi Gauss-Lobatto, and Jacobi Gauss along with $x=\pm 1$. To facilitate practical application, we derive the expression for the pseudo-spectral derivative matrix for the Jacobi Gauss nodes on a closed interval, considering the presence of the staggered grid. Additionally, we provide numerical evidence demonstrating the spectral convergence of these schemes in both space and time.