1. Semi-Lagrangian Approximation in the Time-Dependent Navier-Stokes Equations
Vladimir V. Shaydurov
Institute of Computational Modeling of Siberian Branch of Russian Academy of Sciences, Krasnoyarsk
Beihang University, Beijing
in cooperation with G. Shchepanovskaya and M. Yakubovich

2. Contents Convection-diffusion equations:
Modified method of characteristics.
Conservation law of mass:
Approximation in norm.
Finite element method:

3. Pironneau O. (1982): Method of characteristics
The main feature of several semi-Lagrangian approaches consists in approximation of advection members
as one “slant” (substantial or Lagrangian) derivative
in the direction of vector

4. Pointwise approach in convection-diffusion equation
The equation with this right-hand side is self-adjoint.

5. Approximation of slant derivative
Apply finite element method at the time level and use appropriate quadrature formulas for the lumping effect
Two ways for approximation of slant derivative
1. Approximation along vector
2. Approximation along characteristics (trajectory)

6. Approximation of substantial derivative along trajectory
Solution smoothness usually is better along trajectory
Asymptotically both way have the same first order of approximation

7. Finite element formulation at time level
Intermediate finite element formulation
Final formulation

8. Interpolation-1 Stability in norm:
Chen H., Lin Q., Shaidurov V.V., Zhou J. (2011), …

9. Interpolation-1 Stability in norm and conservation law:
impact of four neighboring points into the weight of

10. Interpolation-2

11. Connection between interpolations

12. Improving by higher order differences

13. Solving two problems with the first and second order of accuracy

14. Navier-Stokes equations. Computational geometric domain

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

16. Notation

17. Notation

18. Notation

19. Initial and boundary conditions

21. Boundary conditions at outlet supersonic and rigid boundary

22. Boundary conditions at subsonic part of computational boundary
a wake

23. Direct approximation of
Curvilinear hexahedron V:
Trajectories:

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

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

26. Discrete approach

27. Matrix of finite element formulation at time layer

28. Supersonic flow around wedge
M=4, Re=2000
angle of the wedge β ≈ 53.1º, angle of attack  = 0º
Density and longitudinal velocity at t = 8
Density and longitudinal velocity at t = 20

29. Density and longitudinal velocity at t = 50

30. Supersonic flow around wedge for nonzero angle of attack
M=4, Re=2000
angle of the wedge β = 53.1º, angle of attack  = 5º
Density and longitudinal velocity at t = 6
Density and longitudinal velocity at t = 8

31. Density and longitudinal velocity at t = 10

32. Density
M=4, Re=2000
Angle of the wedge β ≈ 53.1°
Longitudinal velocity

33. Conclusion Conservation 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
The better smooth properties and the better approximation along trajectories

34. Thanks for your attention!