96 / 12 / 20
NCHC
The Development and Application of the Space-Time
Conservation Element and Solution Element Method
T.-I Tseng
National Center for High-performance Computing
Institute of Physics, NCTU
Dec 20, 2007
Background of the Space-Time CE/SE Method
2
Transport Theory
 Mass
conservation in
continuum
mechanics
 Transport theory
d
dV 0

dt V ( t )

V t dV  S (V ) u  ds  0

   u   0
t
z
y
x
3
The Finite Volume Method
 The rate of change of

conserved properties
in a finite volume (FV)
is equal to its flux
across cell boundaries
The FV methods focus
on calculating spatial
flux (a temporal
evolving spatial flux).

V t dV   S (V ) u  ds
 dV
t2
t1
V
t2
   dt
t1
 u  ds
S (V )
FV
t
x
4
The Space-Time CE/SE Method
 The convection


equation is a spacetime divergence free
condition
Let x1 = x and x2 = t be
the coordinates of a 2D
Euclidean space E2
Using Gauss’
divergence theorem,
one obtains the spacetime flux balance
equation
 ( u ) 

0
x1
x2

  u 
  h  0, where h   
 

   h dV  0
R
 
 h  ds  0
s(R)
t
ds
R
dr
r+dr
r
S(R)
x
5
The Space-Time CE/SE Method
 The Euler equations

are three space-time
divergence free
conditions
Using Gauss’
divergence theorem,
one obtains the spacetime flux balance
equation
U F

0
x2 x1
u 
 u1    
 f1  
   
  

2
U   u 2    u  F   f 2    u  p 
 u   e 
 f   ( e  p )u 
 3  
 3 


  h m  0, m  1,2,3

h m  ( f m , u m )T

   h m dR  0
R


h

d
s
 m 0
S (R)
6
Space-Time CE/SE Method


the discretization step
Space time region is divide into
non- overlapping Conservation
Elements (CEs) and Solution
Elements (SEs).
A staggered space-time mesh
7
The Solution Element


Flow properties are assumed continuous
inside each SE.
The 1st order Taylor series expansion is
used inside a SE
(j,n)
U ( x, t; j , n )  U  ( U ) ( x  x j )  ( U ) (t  t )
*
n
j
n
x j
n
t j
n
F* ( x, t; j, n )  Fjn  (Fx )nj ( x  x j )  (Ft )nj (t  t n )

Inside a SE,
Solution element; SE(j, n)
(Ut )nj  (Fx )nj   A nj (U x )nj
(j,n)
(Ft )nj   A nj (Ut )nj   A nj A nj (U x )nj

U and Ux are the unknowns to be solved;
all other properties can be expressed by
them.
(j-1/2,n-1/2)
(j+1/2,n-1/2)
8
The Conservation Element
B
A
F
(j,n)

The space-time region is divided into
non-overlapping CEs
The space-time flux conservation is
imposed over CE- and CE+

C

(s )
x
t
t 2
n 1/ 2
n 1/ 2

(umx ) j 1/ 2 
( f m ) j 1/ 2 
( f mt ) nj11// 22 , m  1,2,3
4
x
4x
CE+
CEC
D
D
E
(j+1/2,n-1/2)
(j-1/2,n-1/2)
Basic Conservation element; BCE(j, n)
t
1
(um ) nj  [(um ) nj 11// 22  (um ) nj1/12/ 2  ( sm ) nj11// 22  ( sm ) nj11// 22 ], m  1,2,3
2
n 1/ 2
m j 1/ 2
(j,n)
(j,n)
S (V )
For one conservation equation, CEand CE+ provide two conditions.
For the 1D Euler equations, CE- and
CE+ provide 6 conditions for the 6
components of U and Ux at point (j, n)
E
(j+1/2,n-1/2)
(j-1/2,n-1/2)
Conservation element; CE(j, n)
A
F
B
A


h

d
s
 m 0

D
(j,n)
x
(j-1/2,n-1/2)
CE-
CE+
(j+1/2,n-1/2)
9
2D Space-Time Mesh
 Triangular unstructured mesh
 Quad cylinders for CEs
 3 CEs between A, E, C and G’, for the 3
unknowns U, Ux and Uy.
G’


 hm  ds  0
S (V )
D
E
G
C
t
y
F
x
A
Time marching
E
B
G
y
x
C
A
10
The CE and SE in 2D


Three CEs: Quadrilateral cylinder EFGDEFGD(1)
CDGBCDGB(2), and ABGFABGF ( 3 ).
One SE: Four planes ABCDEF + GGBB + GGDD
+ GGFF + their immediate neighborhood.
B''
A
B
CE(2)
C
A
n
F
G
t
B
CE(3)
D''
t
CE(1)
D
n+1/2
F''
G''
G
F
n
F'
n-1/2
C
E
A'
B'
B'
F'
C'
n-1/2
D
E
G'
G'
D'
E'
D'
11
3D Space-Time Mesh




Tetrahedrons are used as the
basic shape
Every mesh node has 4
neighboring nodes
The projection of a spacetime CE on the 3D space is a
6-surface polygons
Flux conservation over 4 CEs
determine the 4 unknowns: U,
Ux, Uy, and Uz
z
y
x
12
Special Features of the CE/SE Method







Space and time are unified and treated as a single entity.
Separation of conservation element and solution element.
No flux function or characteristics-based techniques and
no reconstruction step.
Numerical dissipation doesn’t overwhelm physical
dissipation
Use the simplest mesh stencil -- triangles for 2D and
tetrahedrons for 3D.
1,2, and 3 D Euler/NS codes for structured/unstructured
meshes running on serial and parallel platforms.
Many application in aero acoustics and combustion.
13
A Space-Time CE/SE Method with Moving Mesh Scheme for
One-Dimensional Hyperbolic Conservation Laws
14
Motivating idea
 A suitable computational grid is important for



solving the hyperbolic systems.
Major challenge of numerical scheme is to capture
the discontinuous solution with sufficient accuracy.
The characteristic of the discontinuous is nonstationary and consequently the fixed uniform grid
may not be the best suited.
The idea of an adaptive grid is to add, remove, or
move the grid concentrated to enhance accurate
and achieve efficiency.
15
Adaptive Mesh

Adaptive Mesh Refinement (AMF):
automatic refinement or coarsening of the spatial mesh.

Adaptive Mesh Redistribution (AMF):
relocates the grid points with a fixed number of nodes.
it’s also known as moving mesh method (MMM).
 Key ingredients of the moving mesh method include:




Mesh equation
Monitor function
Interpolations
n+1/2
n’
n
Interpolation Free MMM :
Interpolation of dependent variables from the old mesh to the new
mesh is unnecessary.
16
Mesh Equation
The logical and physical coordinates:

 
x  ( x1 , x2 ,....., xd )  x ( )

 
  (1 , 2 ,....., d )   ( x )
Equidistribution principles

x j 1
xj

(u( x, t )) dx 
x ( , t )
0
1 xR
(u( x, t )) dx, J  0,1,....., JM  1
JM x L
xM
( x, t )dx    ( x, t )dx  (t )
0
(x )  0 (Quasi-Static equidistribution principles;QSEPs)
x:
:
17
Monitor Function
Scaled solution arc-length
 j 1 / 2  1   2 (ux )2j 1 / 2
where  is a scaling parameter and the cell average of the
solution gradient over the interval [xj, xj+1]
( u x ) j 1 / 2 
 = 0, uniform mesh

x j 1
xj
u x dx
( x j 1  x j )
 >> 1, adapted grid
Smoothing the monitor function
j p
~( x, t ) 
r
  ( r  1)
k  j p
j p

k  j p
k j
k
(
r k j
)
r 1
18
MMCESE
Moving mesh strategy:
use the Gauss-Seidel iteration to solve the mesh equation
v
v
v
v
~
~

j 1 / 2 ( x j 1  x j )   j 1 / 2 ( x j  x j 1 )  0
Moving mesh CE/SE (MMCESE)method
n
x j-1n+1/2
B
A
C
x j-1n
x jn+1/2
n+1/2
x j-1/2
wj-1/2
D
n+1/2
x j+1/2
n+1/2
x j+1
F
E
x jn
wj+1/2
n
x j+1
19
MMCESE Algorithm





Step 1: Given a uniform partition of logical and physical domains, then
specified the initial conditions.
Step 2 : Calculate the monitor function.
Step 3 : Move the grid point by mesh equation.
Step 4 : Evolve the underlying PDEs by CE/SE method on the new
mesh system to obtain the flow variables at new time level.
Step 5 : If tn+1 < T, go to Step 2.
20
Burger Equation
1
ut  ( u 2 ) x  0
2
Initial condition :
1
u( x,0)  sin( 2x )  sin( x )
2
Monitor function :
 j 1 / 2  1   2 (ux )2j 1 / 2
2
1
t = 0.35 Sec.
t = 0.85 Sec.
1.5
2
1.5
t = 0.35 Sec.
t = 0.85 Sec.
Exact solution
1.5
MMCESE solution
1
0.8
Original CESE solution
1
1
0.5
0.6
U
U
0
U
0.5
Time
0.5
0
0
0.4
-0.5 -0.5
-0.5
0.2
-1
1.5
2
-1.5
-1
-1
0
0.5
1
X
1.5
2
0
0
0.5
1
X
1.5
2
-1.5
-1.5
0
0
0.5
0.5
1
X
1
X
1.5
1.5
2
2
SE, (a) trajectory of Mesh. (b) comparison between moving (symbols) Fig. 2. Solution of MMCESE, (a) trajectory of Mesh. (b) comparison between moving (symbols)
h scheme solutions.
and fixed (lines) mesh scheme solutions.
21
Sod Problem
L
H
PH
Initial conditions :
Monitor function :
1.2
PL
  , v, p   1.0,0,1.0, x  0

 , v, p   0.125,0,0.1, x  0
  1   2 [(  x ) /(  x )max ]2
1.2
1.2
0.2
1
0.9
Analytical solution
0.9
0.15
density
Density
Time
0.6
MMCESE solution
0.8
0.1
Original CE/SE solution
0.6
0.6
density
0.5
0.4
0.4
0.3
0.3
D
pressure
0
0.25
0.5
0.2
velocity
-0.5
-0.25
00
0
pressure
0.3
0.2
0.05
0.25
0.5
X
roblem at t = 0.2. (a) trajectory of mesh. (b) flow variables distribution
s denote the moving and fixed grid solutions).
0
-0.5
-0.2
-0.25
0
X
0.25
0.5
velocity
0.1
0.1
-0.5 -0.4
0.15
0.2
0.25
0.3
0.35
0.4
X
-0.2
-0.25
00
X
0.2 0.25
0.4
0.5
X
Fig. 3. Solutions of Sod problem at t = 0.2. (a) trajectory of mesh. (b) flow variables distribution
(dots and solid lines denote the moving and fixed grid solutions).
22
Piston Problem
Moving Boundary
v
v
v
v
~
~

j 1 / 2 ( x j 1  x j )   j 1 / 2 ( x j  x j 1 )  0
t
t
x(0)  x00   vbl dt,
0
x( JM )  xJM
  vbrdt
0
(a)
(b)
0.3
0.25
0.25
1.4
0.25
0.2
Time
0.2
0.2
Piston Locus
0.15
1.3
t = 0.06
0.15
0.15
Time
T
Simple
compression waves
0.05
Density
0.1
0.1
0
0
0
1
2
X
3
4
t = 0.14
1.2
0.1
5
t = 0.1
t = 0.02
1.1
0.05
0.05
Tail
1
0
Head
0 0
0
1
2
2.5
X
3
4
X
5
5
0
1
2
3
4
5
X
Fig. 4. Compression wave generated in a tube by a piston. (a) density distribution.
(b) trajectory of mesh.
23
Conclusion
 MMCESE not only maintains the essential



features of the original CE/SE method but also
clusters the mesh space at the locations where
large variation in physical quantities exists.
Current approach can be extend to moving
boundary problem easily.
MMCESE is an interpolation free MMM.
Computation accuracy and efficiency can be
improved by this approach.
Tseng, T.I, and Yang R.J.. “A Space-Time Conservation Element and Solution Element Method with Moving Mesh Scheme for OneDimensional Hyperbolic Conservation Laws,” the 6th Asian Computational Fluid Dynamics Conference, 2005.
24
Applications in Shallow Water Equations
25
Shallow Water Equations
Depth averaging of the free surface flow equations under the
shallow-water hypothesis leads to a common version of the
shallow-water equations (SWEs)
um f m

 S m , m  1,2
t
x
0
hv


h



,
, S m  
um   , f m   2
2

 hv 
 hv  gh / 2 
 gh( s0  s f ) 
where s0 and sf are bed slop and friction slop, respectively.
s0  Z
The friction slop is determined by the Manning formula
sf 
n 2 (hu ) hu
h10 / 3
n is the Manning roughness coefficient.
26
CE/SE method with Source terms
By using Gauss’ divergence theorem in space-time region, the
SWEs can be written in integral form
 h  ds   S (u )dV
For any point belong the solution element
S (V )
m
V
m
m

um ( x, t; j, n)  (um ) nj  ( x  x j )(umx ) nj  (t  t n )(umt ) nj

f m ( x, t; j, n)  ( f m ) nj  ( x  x j )( f mx ) nj  (t  t n )( f mt ) nj
From SWEs, one can get
(umt )nj  ( f mx )nj  Sm ((um )nj )
n
As a result, there are two independent marching variables (um ) j
and (umx ) nj associated with in each solution elements.
Furthermore


h ( x, t; j, n)  ( f m ( x, t; j, n), um ( x, t; j, n))
27
CE/SE method with Source terms
We employ local space-time flux balance over conservation
B
A
F
element to solve the unknowns, i.e.,
(j,n)

S ( CE ( j , n ))
hm  ds  
CE ( j , n )
S m (um )dV
C
D
(j-1/2,n-1/2)
E
(j+1/2,n-1/2)
because the boundary of CE(j, n) is a subset of the union of
SE(j, n), SE(j-1/2, t-1/2), and SE(j+1/2, t-1/2), the
conservation laws imply that
(um ) nj 
t
1
S m ((um ) nj )  [(um ) nj 11// 22  (um ) nj1/12/ 2  ( sm ) nj11// 22  ( sm ) nj11// 22 ], m  1,2
2
2
n 1/ 2
m j 1/ 2
(s )
x
t
t 2
n 1/ 2
n 1/ 2

(umx ) j 1/ 2 
( f m ) j 1/ 2 
( f mt ) nj11// 22 , m  1,2
4
x
4x
The space derivatives of flow variables are evaluated
using the α scheme.
Yu, S.T., and Chang, S.C. “Treatments of Stiff Source Terms in Conservation Law by the Method of Space-Time Conservation
Element and Solution Element,”, AIAA Paper 97-0435,1997
28
1D Dam-Break Problem
with Finite Downstream Water Depth
Two different initial water depths are assigned to the upstream and downstream parts
of a horizontal, frictionless, infinitely wide rectangular channel including a dam. The
upstream water depth is 100 m and the downstream one is 1 m. Spatial domain is
2000 m length and it is discretized with 200 elements.
100
Analytical solution
Numerical result
80
H
60
40
20
CFL 
0
-1000
-500
0
C t
x
500
1000
X
29
Propagation and Reflection
It is assumed that a dam located in the middle of a 1000 m long channel separates
reservoir with depth hr and tailwater depth ht. At the ends of the channel are the walls
that the wave cannot surmount. The boundary conditions with free-slip and zero
discharge are satisfied at the ends of the channel. At time t = 30 s and t = 75 s after
dame break, the results with different ratios of water depth (R = 0.15 and 0.001)
10
10
T = 30 sec.
T = 75 sec.
T = 30 sec.
T = 75 sec.
8
8
6
H
H
6
4
4
2
2
0
0
0
200
400
600
X
800
1000
0
200
400
600
800
1000
X
30
Propagation and Reflection
It is assumed that a dam located in the middle of a 1000 m long channel separates
reservoir with depth hr and tailwater depth ht. Comparisons the numerical results with
and without the bed friction at time t = 30 s after dame with bed slop and different
ratios of water depth (R = 0.15 and 0.001).
12
12
n = 0.03, S0 = 0.003
n = 0, S0 = 0.003
8
8
H
10
H
10
n = 0.03, S0 = 0.003
n = 0, S0 = 0.003
6
6
4
4
2
2
0
0
200
400
600
X
800
1000
0
0
200
400
600
800
1000
X
31
Interaction of 1D Bore Waves
It is supposed that in two 1000 m long channels there are two dams located at 300 m
and 600 m, respectively. Each channel is divided into three parts by both dames. The
initial clam water depths are h01, h02, and h03.
(1) h01 = 20 m, h02 = 3 m, and h03 = 10 m;
(2) h01 = 20 m, h02 = 10 m, and h03 = 2 m;
25
25
T = 8 sec.
T = 24 sec.
T = 10 sec.
T = 30 sec.
15
15
H
20
H
20
10
10
5
5
0
0
200
400
600
X
800
1000
0
0
200
400
600
800
1000
X
32
Steady flow over a bump with hydraulic jump
A steady-state transcritical flow over a bump, with a smooth transition followed by a
hydraulic jump is simulated. The channel is infinitely large, horizontal, frictionless,
25 m long.
2
3


0
.
2

0
.
05
(
x

10
)
,
8

x

12
q(
0
,t)

0
.
18
m
/s
z
(
x
)



,
otherw
h(
0
,t)

0
.
33
m
 0

H
Bed level
0.4
H, Z
0.3
0.2
0.1
0
5
10
15
20
25
X
33
2D Dam-Break Problem
10
8
D
A square box of 200╳200 m2 with a
horizontal bed is divided into two equal
compartments. The initial still water depth is
10 m on one side and 5m and 0.01m on the
other side of the dividing wall for the wet
bed and dry test cases, respectively. The
breach is 75 m in length, and the dame is 15
m in thickness.
6
0
4
200
50
150
X 100
100
Y
150
50
200
0
200
150
Y
12
100
10
8
4
50
2
0
0
-2
200
50
0
D
6
150
0
50
100
150
X
200
X 100
100
Y
150
50
200
0
Tseng, T.I, and Yang R.J.. “Solution of Shallow Water Equations Using Space-Time Conservation Element and Solution Element
Method,” the 14th National Computational Fluid Dynamics Conference, 2007.
34
35
Acknowledgement
• Dr. S.-C., Chang
NASA Gleen Research Center
• Prof. S.-T., Yu
Ohio State University
• Dr. C.-L., Chang
NASA Langley Research Center
• Prof. R.-J., Yang
NCKU
• Dr. Z.-C., Zhang
Livermore Software Technology Co.
• Prof. W.-Y., Sun
NCTFR
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36
Thanks for Your Attention
37