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 shaidurov04@mail.ru in cooperation with G. Shchepanovskaya and M. Yakubovich

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

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

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

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)

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

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

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

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

Interpolation-2

Connection between interpolations

Improving by higher order differences

Solving two problems with the first and second order of accuracy

Navier-Stokes equations. Computational geometric domain

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

Notation

Notation

Notation

Initial and boundary conditions

Boundary conditions at outlet supersonic and rigid boundary

Boundary conditions at subsonic part of computational boundary a wake

Direct approximation of Curvilinear hexahedron V: Trajectories:

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

Gauss-Ostrogradskii Theorem in the case of boundary conditions:

Discrete approach

Matrix of finite element formulation at time layer

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

Density and longitudinal velocity at t = 50

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

Density and longitudinal velocity at t = 10

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

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

Thanks for your attention!