Local Boundary Condition Based Spectral Collocation Methods
for 2D and 3D Navier-Stokes Equations

We present a simple approach to accurately and efficiently computing local boundary conditions in spectral collocation schemes for the Navier-Stokes equations 2D and 3D dimensions. Access to these local boundary values makes possible the decoupling of the computation of the primary flow variables, resulting in highly efficient schemes. In 2D the local vorticity boundary values are employed in the vorticity-stream function formulation, while in 3D local values of a Neumann boundary condition for the pressure in the velocity-pressure formulation are used. In both cases these boundary conditions are approximated by differentiating a local Lagrange interpolant at the boundary. The straightforward extension of the approach to the Boussinesq system is also discussed. The resulting schemes are well suited for the simulation of moderate to high Reynolds and Rayleigh number flows. Accuracy checks, simulations of the lid-driven cavity flow, of a differentially heated cavity flow, and of a Rayleigh-Bernard convection problem for Rayleigh number up to 10¹⁰ are presented to demonstrate that the schemes are capable of producing accurate results at a reasonable computational cost.