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Semi-Lagrangian Approximation of Navier-Stokes Equations

Semi-Lagrangian Approximation of Navier-Stokes Equations. Vladimir V. Shaydurov Institute of Computational Modeling of SB RAS Beihang University shaidurov04@gmail.com. Contents. Approximation in norm. Modified method of characteristics. Convection-diffusion equation.

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Semi-Lagrangian Approximation of Navier-Stokes Equations

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  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

  16. Finite element formulation at each time level

  17. The channel with an obstacle at the inlet

  18. The component of velocity u The distribution of density

  19. 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

  20. Thanks for your attention!

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