On the accuracy of difference scheme for Navier–Stokes equations

The article presents a study of difference schemes in time, which accuracy can be arbitrarily high. We present difference schemes in time for solving the Navier–Stokes equations, where series expansions are used to find the singularities of solutions of the Euler equations. These methods are generalized in this article for the arbitrary order schemes and for solving the Burgers equation and the Navier–Stokes equations for an incompressible fluid. The impact of the scheme on the calculation accuracy is examined. First, the method is applied to the test case associated with the Burgers equation, and then the problem of three-dimensional incompressible flow finding by solving the Navier–Stokes equations is considered. It is shown that the finite-difference scheme used to calculate the time derivatives is the main source of deviations of the approximate solution from the exact solution.