Weak Versus Strong Wall Boundary Conditions for the Incompressible Navier-Stokes Equations

Abstract

The pressure-velocity formulation of the incompressible Navier-Stokes equations is solved using high-order finite difference operators satisfying a summation-by-parts property. Two methods for imposing Dirichlet boundary conditions (one strong and one weak) are presented and proven stable using the energy method. Additionally, novel diagonal-norm second-derivative finite difference operators are derived with highly improved boundary accuracy. Accuracy and convergence measurements are presented and verified against theoretical expectations. Numerical experiments also show that subtle effects close to solid walls are more efficiently captured with strong boundary condition imposition methods rather than weak (less degrees of freedom required).