A Numerical Study of the Peridynamic Differential Operator Discretization of Incompressible Navier–Stokes Problems

Abstract

We study the incompressible Navier–Stokes equations using the Projection Method. The applications of interest are the classical channel flow problems such as Couette, shear, and Poiseuille. In addition, we consider the Taylor-Green vortex and lid-driven cavity applications. For discretization, we use the Peridynamic Differential Operator (PDDO). The main emphasis of the paper is the performance of the PDDO as a discretization method under these flow problems. We present a careful numerical study with quantifications and report convergence tables with convergence rates. We also study the approximation properties of the PDDO and prove that the N-th order PDDO approximates polynomials of degree at most N exactly. As a result, we prove that the PDDO discretization guarantees the zero row sum property of the arising system matrix.