An optimally accurate discrete regularization for second order timestepping methods for Navier–Stokes equations

We propose a new, optimally accurate numerical regularization/stabilization for (a family of) second order timestepping methods for the Navier–Stokes equations (NSE). The method combines a linear treatment of the advection term, together with stabilization terms that are proportional to discrete curvature of the solutions in both velocity and pressure. We rigorously prove that the entire new family of methods are unconditionally stable and O(Δt2)O(Δt2) accurate. The idea of ‘curvature stabilization’ is new to CFD and is intended as an improvement over the commonly used ‘speed stabilization’, which is only first order accurate in time and can have an adverse effect on important flow quantities such as drag coefficients. Numerical examples verify the predicted convergence rate and show the stabilization term clearly improves the stability and accuracy of the tested flows.