Semi-Lagrangian Approximation of Navier-Stokes Equations

1. Semi-Lagrangian Approximation of Navier-Stokes Equations Vladimir V. Shaydurov Institute of Computational Modeling of SB RAS Beihang University shaidurov04@gmail.com

2. Contents • Approximation in norm. • Modified method of characteristics. • Convection-diffusion equation. • Approximation in norm. • Conservation law of mass. • Approximation in norm. • Conservation law of energy.

3. The main original feature of semi-Lagrangian approach consists in approximation ofall advection members as one “slant”(substantial or Lagrangian) derivative in the direction of vector

4. First example: convection-diffusion equation

5. Approximation of substantial derivative along trajectory approach

6. The equation at each time level became self-adjoint! Pironneau O. (1982), … Chen H., Lin Q., Shaidurov V.V., Zhou J. (2011), …

7. Computational geometric domain

8. Navier-Stokes equations In the cylinder we write 4 equations in unknowns

9. Notation

10. Notation

11. Notation

12. How to avoid the Courant-Friedrichs-Lewy restriction for high Reynolds numberapproach Curvilinear hexahedron V: Trajectories:

13. Due to Gauss-Ostrogradskii Theorem: Approximation of curvilinear quadrangle Q:

14. Gauss-Ostrogradskii Theorem in the case of boundary conditions:

15. approach

20. Finite element formulation at each time level

21. The channel with an obstacle at the inlet

22. The component of velocity u The distribution of density

23. Conclusion • Stability of full energy (kinetic + inner) • Approximation of advection derivatives in the frame of finite element method without artificial tricks • The absence of Courant-Friedrichs-Lewy restriction on the relation between temporal and spatial meshsizes • Discretization matrices at each time level have better properties (positive definite) • The better smooth properties and the better approximation along trajectories

24. Thanks for your attention!