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.

• Approximation in norm.

Conservation law of mass.

• Approximation in norm.

Conservation law of energy.

L

1L

2L

• The main original feature of semi-Lagrangian approach consists in approximation of all advection members

as one “slant” (substantial or Lagrangian) derivative

in the direction of vector

u vdt y

d

t x

d

d l

.l

First example: convection-diffusion equation

u v ft x y

d

d lf

1

ˆˆ ˆ( , , ),

.ˆ

ˆ ˆ( , , ),k k

dxu t x y

dt t t tdy

v t x ydt

Approximation of substantial derivative along trajectory approachL

( )tf

• The equation at each time level became self-adjoint!

Pironneau O. (1982), …Chen H., Lin Q., Shaidurov V.V., Zhou J. (2011), …

Computational geometric domain

out

inrigid

Navier-Stokes equations

(0, )fint , , ,u v eIn the cylinder we write 4 equations in unknowns

( )0,xyxxD u P

Dt x x y

( )0,xy yyD v P

Dt y x y

( ).yx

qqD e u vP

t x y x y

( ) ( ) 0u vt

D

t yD x

Notation

( , , ) is density;t x y

( , , ), ( , , ) are components of the vector velocity ;u t x y v t x y u

( , , ) is internal energy of mass unit;e t x y

( , , ) is pressure;P t x y

1( , , ) ( , , )

Ret x y t x y

is the dynamic coefficient of viscosity:

21 , 0.76 0.9M e

Notation

, , are the components of the stress tensor (matrix)

:

xx xy yy

xx xy

xy yy

2 22 , 2 ,

3 3

xx yy

xy

u v v u

x y y x

u v

y x

Notation

, are components of the vector density of a heat flow:x yq q

( , , ) , ( , , )Pr Prx y

e eq t x y q t x y

x y

( 1) .P e

2 2 222 2

.3 3

u v v u u v

x y x y x y

Re is the Reynolds number; Pr is the Prandtl number;

M is the Mach number; is a gas constant

The equation of state has the following form (perfect gas):

The dissipative function is taken in the form:

How to avoid the Courant-Friedrichs-Lewy restriction for high Reynolds number

approach

( ) ( ) 0u vt

D

t yD x

Curvilinear hexahedron V:

1

ˆˆ ˆ( , , ),

.ˆ

ˆ ˆ( , , ),k k

dxu t x y

dt t t tdy

v t x ydt

Trajectories:

1L

0V Q

D

DV d

td dQ

Approximation of curvilinear quadrangle Q:

1 1, ,

4, , 2

1( )

k ki j i j

hk i jQ Q

d dQ O h dQh

Due to Gauss-Ostrogradskii Theorem:

0V Q R

dV d dQ u dDt

RD

1 1, ,

1 1, ,

4

, , 2 2

( )

1 1

k ki j i j

k ki j i j

Q R

hk i j Q R

d dQ u dR O h

dQ u dRh h

Gauss-Ostrogradskii Theorem in the case of boundary conditions:

( )0xyxxD u P

Dt x x y

( , ) :x y

( , , ) : ( , , ) ( , ),

0

kt x y t x y x y

u vy

d

xdt t

approach

2L

( )0xyxxD u P

Dt x x y

( ) ( ) ( )D u D u D u du

Dt Dt Dt dt

( )0xyxxD u P

Dt x x y

( )0xyxx

V

D u PdV

Dt x x y

1k k

xy yy

t t t tQ

v P vd dQ

y x y

1k k

xyxxt t t t

Q

u P ud dQ

x x y

( ) 10

2yxqD q u v

PDt x y x y

( ) yxqD e q u v

PDt x y x y

1 2e

2( ) yxqD q u v

PDt x y x y

2 22( ) ( ) ( )

2D D D D

Dt Dt Dt Dt

( )0

2yx

V

qD q u vP dV

Dt x y x y

1

1

2 k k

yxt t t t

Q

qq u vP d dQ

x y x y

1k k

xyxxt t t t

Q

u P ud dQ

x x y

1k k

xy yy

t t t tQ

v P vd dQ

y x y

1

1

2 k k

yxt t t t

Q

qq u vP d dQ

x y x y

Finite element formulation at each time level

kt t

The channel with an obstacle at the inlet

The component of velocity u

The distribution of density

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

• Thanks for your attention!