A Strongly-Consistent Difference Scheme for 3D Nonlinear Navier-Stokes Equations

This paper constructs a strongly-consistent explicit finite difference scheme for 3D constant viscosity incompressible Navier-Stokes equations by using of symbolic algebraic computation. The difference scheme is space second order accurate and temporal first order accurate. It is proved that difference Grobner basis algorithm is correct. By using of difference Grobner basis computation method, an element in Grobner basis of difference scheme for momentum equations is a difference scheme for pressure Poisson equation. The authors find that the truncation errors expressions of difference scheme is consistent with continuous errors functions about modified version of above difference equation. The authors prove that, for strongly consistent difference scheme, each element in the difference Grobner basis of such difference scheme always approximates a differential equation which vanishes on the analytic solutions of Navier-Stokes equations. To prove the strongly-consistency of this difference scheme, the differential Thomas decomposition theorem for nonlinear differential equations and difference Grobner basis theorems for difference equations are applied. Numerical test certifies that strongly-consistent difference scheme is effective.