Semi-Lagrangian Approximation in the TimeDependent Navier-Stokes Equations
Vladimir V. Shaydurov
Institute of Computational Modeling of Siberian Branch of
Russian Academy of Sciences, Krasnoyarsk
Beihang University, Beijing
[email protected]
in cooperation with G. Shchepanovskaya and M. Yakubovich
Contents
• Convection-diffusion equations:
Modified method of characteristics.
• Conservation law of mass:
Approximation in l1  norm.
• Finite element method:
Approximation in l2  norm.
• Pironneau O. (1982): Method of characteristics
The main feature of several semi-Lagrangian approaches
consists in approximation of advection members
d  



u
v
dt
t
x
y
as one “slant” (substantial or Lagrangian) derivative
d
dl
in the direction of vector l
Pointwise approach in convection-diffusion equation



u
v
   f
t
x
y
d
   f
dl
d
   f1  f 
in ,   0 on 
dl
The equation with this right-hand side is self-adjoint.
Approximation of slant derivative
Apply finite element method at the time level t k and use
appropriate quadrature formulas for the lumping effect
( Ah  h )( xi , y j )  f1 ( xi , y j ) in h ,  h  0 on h
Two ways for approximation of slant derivative d  d l
1. Approximation along vector l
d
1
(tk , xi , y j )    (tk , xi , y j )   (tk 1, xˆi , yˆ j ) 

dl
2. Approximation along characteristics (trajectory)
 dxˆ
 dt  u (t , xˆ, yˆ ),

 dyˆ  v(t , xˆ, yˆ ),
 dt
tk  t  tk 1.
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 t k
Intermediate finite element formulation
( Ah  h )(tk , xi , y j )  f1 (tk , xi , y j ) in h ,  h (tk , xi , y j )  0 on h
d
f1  f 
dl
Final formulation
1
1
 h (tk , xi , y j )  ( Ah  h )(tk , xi , y j )  f (tk , xi , y j )   h (tk 1, xˆi , yˆ j ) in  h ,


 h (tk , xi , y j )  0 on  h
Interpolation-1
( xˆ , yˆ )
yj
1  ( xˆ  xi ) h ,  2  ( yˆ  y j ) h
xi
u ( xˆ, yˆ )   1 2u ( xi 1 , y j 1 )  (1   1 ) 2u ( xi , y j 1 )
 1 (1   2 )u ( xi 1 , y j )  (1   1 )(1   2 )u ( xi , y j )
Stability in l  norm:
u ( xˆ, yˆ )  max u ( x, y)
( x , y )h
Chen H., Lin Q., Shaidurov V.V., Zhou J. (2011), …
Interpolation-1
1  ( xˆ  xi ) h ,  2  ( yˆ  y j ) h
( xˆ , yˆ )
yj
xi
Stability in l1  norm and conservation law:
impact of four neighboring points into the weight of u ( xi , y j )
 1 2u ( xi , y j )  (1   1 ) 2u ( xi , y j )
 1 (1   2 )u ( xi , y j )  (1   1 )(1   2 )u ( xi , y j )  (1  c )u ( xi , y j )
Interpolation-2
( xˆ , yˆ )
yj
1  ( xˆ  xi ) h ,  2  ( yˆ  y j ) h
xi
u 2 ( xˆ, yˆ )   1 2u 2 ( xi 1, y j 1 )  (1   1 ) 2u 2 ( xi , y j 1 )
 1 (1   2 )u 2 ( xi 1 , y j )  (1   1 )(1   2 )u 2 ( xi , y j )
Connection between interpolations
1  ( xˆ  xi ),  2  ( yˆ  y j )
u 2 ( xˆ, yˆ )  u 2 ( xˆ, yˆ )
  u ( x
1
2
i 1
, y j 1 )  (1   1 ) 2u ( xi , y j 1 )
 1 (1   2 )u ( xi 1 , y j )  (1   1 )(1   2 )u ( xi , y j ) 
2
  1 2u 2 ( xi 1 , y j 1 )  (1   1 ) 2u 2 ( xi , y j 1 )
 1 (1   2 )u 2 ( xi 1 , y j )  (1   1 )(1   2 )u 2 ( xi , y j )  
 1 2  (1   1 ) 2 )  1 (1   2 )  (1   1 )(1   2 ) 
Improving by higher order differences
d
1
(tk , xi , y j )    (tk , xi , y j )   (tk 1, xˆi , yˆ j ) 

dl
d
1
(tk , xi , y j )    (tk 1 , xi , y j )   (tk 1, xˆi , yˆ j ) 
2
dl

 (t , x , y )   (t



1
 (tk 1 , xi , y j )  2  (tk , xi , y j )   (tk 1, xˆi , yˆ j ) 

2
1
k
i
j
k 1
, xˆi , yˆ j ) 
Solving two problems with the first and second order of
accuracy
k  1,..., m
1
1
 h (tk , xi , y j )  ( Ah  h )(tk , xi , y j )  f (tk , xi , y j )   h (tk 1 , xˆi , yˆ j ) in  h ,


 h (tk , xi , y j )  0 on  h
k  1,..., m
1
1
 2h (tk , xi , y j )  ( Ah  2h )(tk , xi , y j )  f (tk , xi , y j )   2h (tk 1 , xˆi , yˆ j )


1
   h (tk 1 , xi , y j )  2  h (tk , xi , y j )   h (tk 1 , xˆi , yˆ j )  in  h ,
2
 2h (tk , xi , y j )  0 on  h
Navier-Stokes equations. Computational geometric domain
 out
in
rigid
 out

 out
Navier-Stokes equations
In the cylinder (0, t fin )  
we write 4 equations in unknowns
d
u
v

 0
dt
x
y
du P  xx  xy




 0,
dt x
x
y
dv P  xy  yy
 


 0,
dt y
x
y
 u v 
de qx q y
 

  P     .
dt x
y
 x y 
 , u, v, e
Notation
 (t , x, y) is density;
u(t , x, y), v(t , x, y) are components of the vector velocity u;
e(t , x, y) is internal energy of mass unit;
P(t , x, y) is pressure;
 (t , x, y) 
1 
 (t , x, y)
Re
  is the dynamic coefficient of viscosity:
      1  M


2 

e , 0.76    0.9
Notation
 xx , xy , yy are the components of the stress tensor (matrix)
 xx  xy 

:


yy 
 xy
2  u
v 
2  v
u 
 xx    2   ,  yy    2   ,
3  x y 
3  y x 
 u
v 
 xy     
 y x 
Notation
 q , q  are components of the vector density of a heat flow:
x
y
qx (t , x, y)  

Pr

e
 e
, q y (t , x, y)   
x
Pr 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):
P  (  1)  e.
The dissipative function is taken in the form:
 2  u 2 2  v 2  v u 2  u v  2 
                 .
 3  x  3  y   x y   x y  


Initial and boundary conditions
t  0, x  :
 (0, x)  0 (x)  0
u(0, x)  u0 (x)
e(0, x)  e0 (x)  0
t, x   (0, T ) in :
 (t, x)  in (t, x)  0
u(t, x)  uin (t, x)
e(t, x)  ein (t, x)  0

2
2

e

(
u

v
) 2 d 




t
    e  ( u 2  v 2 ) 2  ( u  n) d   


 Pu  u  q   n d 
 u v 
de qx q y
 

 P 
 
dt x
y
 x y 
 e :
12
 u v 
d 2 qx q y



 P 
 
dt
x
y
 x y 
d 2
d
 2
dt
dt

 u v 
d 1  qx q y



 P       0

dt 2  x
y
 x y 

Boundary conditions at outlet supersonic and
rigid boundary
t, x [0, T ] super :
 du
 dt  0

 dv
 0
 dt
 d
 dt  0

t, x  [0, T ]  rigid :
u0

0
n
Boundary conditions at subsonic part of
computational boundary
t, x [0, T ] sub :
Pn  n  Pout n

0
n
a wake
Direct approximation of
D  


 (  u )  (  v)  0
Dt t x
y
Curvilinear hexahedron V:
Trajectories:
 dxˆ
 dt  u (t , xˆ, yˆ ),

 dyˆ  v(t , xˆ, yˆ ),
 dt
tk  t  tk 1.
Due to Gauss-Ostrogradskii Theorem:

V
D
dV  0 
Dt


 d     dQ
Q
Approximation of curvilinear
quadrangle Q:


 d  
Qik, j 1
 dQ  O( h3 )  kh,i , j 
1
 dQ
2 Q k 1
h i, j
Gauss-Ostrogradskii Theorem in the case of boundary conditions:

V
D
dV  0 
Dt




 d  
 
Qik, j 1
h
k ,i , j
 d     dQ   u dR
Q
 dQ  
Rik, j 1
R
 u dR  O( h3 )
1
1
 2  k 1  dQ  2  k 1  u dR
h Qi , j
h Ri , j
Discrete approach

 ( x, y ) :
du P  xx  xy



0
dt x
x
y
Matrix of finite element formulation at time layer t  tk











D1  A11
A12
A13
A21

D2  A22

A23
A31
A32

D3  A33

O
O
O




O



O



D4  A44 


O
Supersonic flow around wedge
M=4, Re=2000
angle of the wedge β ≈ 53.1º, angle of attack  = 0º
hx  0.01, hy  0.005, (300  400),   0.0005
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 and longitudinal velocity at t = 20
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!