# Time reversed algorithm for pure convection

Mécanique

22 févr. 2014 (il y a 4 années et 2 mois)

69 vue(s)

Time reversed algorithm for pure
convection

V
.М.
Goloviznin

Mathematical modeling transport phenomena on
computational grid is one of the fundamental
problems of the modern computational
mathematics

Simplest transport equation is time reversed

0; 0
c c Const
t x
 
 
    
 
Substitution

* *
; ;
t t c c
   
is led to the same equation

* *
*
0; 0
c c Const
t x
 
 
    
 

Finite difference schemes of time
reversed quality

On the regular computational grid in the plane (t,x) are known only
two explicit finite difference schemes of second order of accuracy and
implicit one.

1 1
1 1
0;
2 2
n n n n
i i i i
c
h
   

 
 
 
  
x

t

One of them is well known

Leap
-
Frog scheme

Next one is Iserles scheme

1 1
1 1 1
1
0;
2
n n n n n n
i i i i i i
c
h
     
 
 
  
 
  
   
 
 
x

t

Finite difference schemes of time
reversed quality

Implicit time reversible scheme is also well

known Sn Karlsons
scheme

x

t

1 1 1 1
1 1 1 1
1
0;
2 2
n n n n n n n n
i i i i i i i i
c
h h
       
 
   
   
   
   
    
   
   
Leap
-
Frog scheme is transformed into Arakawa

Lilly

Schemes in multidimensional cases and successfully

Explored in ocean modeling.

Sn

Karlson scheme in the form of dSn
-
scheme is used

In neutrons transport calculation for nuclear reactor.

Explicit Iserles scheme is transformed into “CABARET”

schemes, witch have a wide sphere of usability.

“CABARET” scheme

Iserles scheme can be rewrite as

1/2 1/2
1/2 1
1/2 1/2 1
1/2 1
1
0;
2
n n n n
n n n
i i i i
i i i
c
h
   
  

 
 
  
 
 
     
Variables will be called as “fluxes variables”.

Variables will be noted as “conservative values”

n
i

x

t

1/2
1/2
n
i

The next step of transform gives “two layers form”

1/2
1/2 1/2 1
1 1/2
1/2 1
1 1/2 1 1
1/2 1/2 1
0;
2
2;
0;
2
n n n n
i i i i
n n n
i i i
n n n n
i i i i
c
h
c
h
   

  
   

  
 
 
   
  
 
  
  
 
  
Dissipation and dispersive surfaces

«
CABARET
»

«
Leap
-
Frog
»

Dissipation

Dispersion

0.2
0.4
0.6
0.8
-2
0
2
0
2
4
6
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
-2
0
2
0
0.5
1
1.5
2
0
0.2
0.4
0.6
0.8
0
0.25
0.5
0.75
1
-2
0
2
0
0.5
1
1.5
2
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
-2
0
2
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
Since the CABARET scheme is second
-
order, according to the Godunov theorem it needs some
procedure to enforcing monotonicity

i i1 n
i i1 n
n
n1 n n n
i1 i1i1/2i
i1/2
xx,x,tt
n
n1 n n n
i1 i1i1/2i
i1/2
xx,x,tt
max max max,,
min min min,,

  

 

  

 
  
 
We constrain the solution so that

n1
i1 i i1 n
n1
i1 i i1 n
max,xx,x,tt
min,xx,x,tt

 

 
  
  
Maximum principle

n1 n1/2 n
i1 i1/2 i
2
 
 
 
Consider 3 values inside 1 cell

n n n
i i 1/2 i 1

 
  
n 1
i 1

Adds on just enough dissipation needed for draining the
energy from unresolved scales, “entropy” condition

New principle item:

direct application of maximum
principle

Main distinguishes CABARET

from upwind leapfrog scheme

CABARET is presented in form of conservation law

CABARET has two type of variables : conservative
-
type and flux
-
type

CABARET is two
-
layers scheme with
very compact, one
-
cell
-
one
-
time
-
level stencil

CABARET is monotonic due to direct application of maximum
principle for flux restriction

I+1

I

n+1

n

Explicit

Stable under 0<CFL<1/d, d=problem dimension

Exact at CFL=0.5, CFL=1

Second
-
order on arbitrary non
-
uniform spatial and
temporal grids

Conservative

Non
-
dissipative

Very compact, one
-
cell
-
one
-
time
-
level stencil

Small dispersion error

Direct application of maximum principle for flux
restriction

Main features of the CABARET
scheme

I+1

I

n+1

n

Another reason to call it CABARET…

Computational stencil of the forerunner of

CABARET scheme

C
ompact
A
ccurately
B
oundary

A
djusting high
-
RE
solution
T
echnique

for Fluid Dynamics

CABARET for gas dynamics flows.

First unique feature.

Gas dynamics:

verification test

Contact discontinuity

Плотность, CFL=0.5, NT=150
0,998
1
1,002
1,004
1,006
1,008
1,01
1,012
1
21
41
61
81
101
Плотность, CFL=0.05, NT=1500
0,998
1
1,002
1,004
1,006
1,008
1,01
1,012
1
21
41
61
81
101
Плотность, CFL=0.5, NT=1500
0
200
400
600
800
1000
1200
1
21
41
61
81
101
Плотность, CFL=0.05, NT=1500
0
200
400
600
800
1000
1200
1
21
41
61
81
101
2
1, 10
LMR LMR
uuuppp


Weak contact discontinuity

Strong contact discontinuity

independence

from amplitude

independence
shock wave thickness from amplitude

0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
0
10
20
30
40
50
60
70
давление
0.999995
1
1.000005
1.00001
1.000015
1.00002
0
10
20
30
40
50
60
70
давление
0.00E+00
2.00E+05
4.00E+05
6.00E+05
8.00E+05
1.00E+06
0
10
20
30
40
50
60
70
давление
Very slow shock wave

Very strong shock wave

Ordinary shock wave

5
10
P
P

6
10
P
P

1
P
P

First Unusual Feature of CABARET
:

Blast Wave problem

0.0 0.1
, 0.1 0.9
0.9 1.0
L
M
R
if x
xt if x
if x

 



 



2 2
3
,,,
1,
0,
10, 10,10.
T
L M R
L M R
L M R
up
u u u
p p p



  
  
  
0
1
2
3
4
5
6
7
0
50
100
150
200
N
потность
0
1
2
3
4
5
6
7
0
500
1000
N
плотность
P,Woodward, P,Colella J,Comp,Phys,,
54
,
115
-
173
(
1984
)

1
-
D shock interaction with density
perturbations: Shu&Osher problem

0
1
2
3
4
5
0
0.2
0.4
0.6
0.8
1
x/L

200
Converged
0
1
2
3
4
5
0
0.2
0.4
0.6
0.8
1
x/L

400
Converged
Shock capturing capability without notable dissipation

Double Mach reflection test

Grid (481x121)

Grid (961x241)

Grid (1921x481)

In a semi
-
open domain an oblique shock wave of Mach
equal to
10
impinges on the horizontal reflective boundary
under an angle of
60
0

Titarev and Toro, 2002; J.Qiu and C.
-
W. Shu, 2003

CABARET for aeroacoustics problems.

Second unique feature
.

D2 acoustic Gaussian pulse propagation

on nonuniform grid

Initial condition

0 0
0
,,(,)
,0
(,)1/,(,)1,(,)0
t t
t
fon fon fon
xy pxy xy
uxy
pxy xy uxy
 

 

 

  

2
2
0 0
(,) exp ( )
pxy xxyy
 


5
00
(,)100,100100,100,
(,)(0,0),10,log2/9
xy
xy

 
  


Computational grid
:

Second Unusual feature of CABARET
:

Acoustic disturbances is not dissipate

D2 acoustic Gaussian pulse propagation

on nonuniform grid

Simulation of vortex flow.

Third unique feature.

2
-
D zero
-
circulation compressible isentropic vortex in a
periodic box

H=1

L=0.05

One revolution: T=1.047

1
1
2 2 2
1
2 2 2
2 2 1
0 0 0 0
( 1)
'1 exp{2 (1 )} 1,'exp{ (1 )}sin,
4
( 1)
'exp{ (1 )}cos,'1 exp{2 (1 )} 1,
4
/;( ) ( );tan (( )/( )),
u
v p p
r L r x x y y y y x x

        


      

 

 
 

 
     
 
 
 
 
 
 
 

 
      
 
 
 
 
 
       
Stationary and stable
solution to EE. But
how long can the
numerical scheme
hold it?

Full Euler equations are solved

Karabasov and Goloviznin, 2008

Single Vortex

Presure

Entropy

Computational grid

50х50

Third Unusual feature of CABARET
:

Stationary vortex is not dissipate

Vortex Dipole

Vortex preserving capability:
2
-
D
zero
-
circulation compressible vortex in a periodic box
domain

t=100

(30x30), 1.5 points per radius (p.p.r.)

(60x60), 3 p.p.r.

(120x120), 6 p.p.r.

Conserves
total k.e.
within ~ 1%

Vortex in a box: stationary
and stable solution to the
Euler equations. But how
long can the numerical
scheme preserve it?

Vorticity

Vortex preserving capability:
what happens with a
conventional
2
nd
-
3
rd

order conservative method? (e.g.,
Roe
-
MUSCL
-
TVD, grid (
240
x
240
))

With the TVD limiter: t=4

With the TVD limiter: t=100

No limiter: t=4

With the limiter the solution is too dissipative

Without the limiter it is too dispersive

(240x240)

Vorticity

Vortex preserving capability

& shock
-
capturing:

Zero
circulation vortex interaction with a stationary normal shock
wave in a wind tunnel: grid (
400
x
200
), density field shown

Weak vortex

Strong vortex

Zhou and Wei, 2003; Karabasov
and Goloviznin, 2007

D2 Backward Step

Re=5000

40 greed point on step

20
greed point on step

10 greed point on step

CABARET for uncompressible flows.

Stream instability on grid (
256
x
256
)

Flow behind turbulizing

grid

Real stream

CABARET simulation

(
256
х
512
)

FLUENT simulation

Submerged jet on computational

grid (
128
x
640
)

Foto

Result of simulation.

Vorticity Field.

Animation

Towards Empiricism
-
Free Large Eddy Simulation
for Thermo
-
Hydraulic Problems

A
animation

15MC,Re=85000

Mixing hydrogen under containment

Remarkable characteristic of CABARET

independence shock wave thickness from amplitude
;

acoustic disturbances is not dissipate.

stationary vortex is not dissipate;

CABARET applicable for lot of challenging problems
:

Transonic aerodynamics

Aeroacoustics

Vortex flow simulation

Ocean modeling

Atmospheric pollution transport

Strongly nonuniform reservoir modeling,

Combustion modeling

Computing turbulent fluid dynamics

Et al

Conclusions

implementation of CABARET in the industry

Innovative scientific problem:

CABARET on new sphere of science and

increasing order of accuracy up to fourth.

Publication

V
.
M
.
Goloviznin

Digital

Transport

Algorithm

for

Hyperbolic

Equations”/

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and

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Numerics

and

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,

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finite
-
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(Preprint

IBRAE

IBRAE

2005
-
01
)
-

М
.:

ИБРАЭ

РАН
,

2005
,

-
37

p
.