The Semi-Lagrangian Method for the Navier-Stokes Problem for an Incompressible Fluid

The two-dimensional time-dependent Navier-Stokes equations are considered for a viscous incompressible fluid in a channel. On the outlet boundary, the modified "do nothing" condition is imposed. To construct a discrete analogue, we use the conforming finite element method in the combination with a semi-Lagrangian approach. The velocity components are approximated by biquadratic elements and the pressure is approximated by bilinear elements on rectangles. To overcome the lack of conservation law of the classical semi-Lagrangian method, we propose its conservative version. To guarantee the energy conservation and the stability in the mean-square norm, we use the discrete analogue of the local integral balance between two neighboring time levels. A numerical experiment shows the convergence of the proposed numerical method.