Involution and Difference Schemes for the Navier–Stokes Equations

Abstract

In the present paper we consider the Navier–Stokes equations for the two-dimensional viscous incompressible fluid flows and apply to these equations our earlier designed general algorithmic approach to generation of finite-difference schemes. In doing so, we complete first the Navier–Stokes equations to involution by computing their Janet basis and discretize this basis by its conversion into the integral conservation law form. Then we again complete the obtained difference system to involution with eliminating the partial derivatives and extracting the minimal Gröbner basis from the Janet basis. The elements in the obtained difference Gröbner basis that do not contain partial derivatives of the dependent variables compose a conservative difference scheme. By exploiting arbitrariness in the numerical integration approximation we derive two finite-difference schemes that are similar to the classical scheme by Harlow and Welch. Each of the two schemes is characterized by a 5×5 stencil on an orthogonal and uniform grid. We also demonstrate how an inconsistent difference scheme with a 3×3 stencil is generated by an inappropriate numerical approximation of the underlying integrals.