An Efficient Error-Controlled POD Reduced-Order Collocation Spectral Method for Unsteady Incompressible Navier–Stokes Equations

Source abstract

The efficient numerical solution of the unsteady incompressible Navier–Stokes equations requires a balance between computational accuracy and efficiency. Although spectral methods provide high-order spatial approximation, the dense algebraic systems generated by their global discretization lead to relatively high computational costs. To address this issue, a POD-based reduced-order collocation spectral method is developed in this paper. First, a nonlinear collocation spectral method (NCSM) is constructed using the projection method and Chebyshev collocation discretization. An SVD-based linearization is then introduced to obtain a linearized collocation spectral method (LCSM) compatible with POD reduction. Based on snapshots generated by the LCSM, a reduced-order collocation spectral method (ROCSM) is further established. Error estimates are derived for the full-order discretization, linearization, and POD reduction, and a residual-based criterion is introduced to guide the selection and updating of the POD basis. Numerical experiments on a body-force-driven multimode vortex flow and an unsteady lid-driven cavity flow verify the theoretical error estimates and demonstrate the effectiveness of the proposed method. The ROCSM accurately reproduces the principal flow structures while substantially reducing the computational cost. Moreover, the results show that the POD-basis updating strategy guided by the POD reduction error effectively prevents error accumulation in long-time simulations. These results confirm that the proposed method provides an accurate and efficient approach for unsteady incompressible Navier–Stokes flows.