the application of the multigrid method in a nonhydrostatic atmospheric model shu-hua chen mmm/ncar
DESCRIPTION
Model Numerical Methods (Semi-implicit scheme) Pressure gradient force and Divergence (Implicit Scheme) Advection (Explicit Scheme ) Eddy diffusion (Explicit scheme) Pressure gradient force and Divergence (Implicit Scheme)TRANSCRIPT
The Application of the Multigrid Method in a Nonhydrostatic
Atmospheric ModelShu-hua ChenMMM/NCAR
Model
Formulae
pR
Cp
100000 Pa
( , , ) ( , , , )( , , ) ( , , , )( , , ) ( , , , )
x y z x y z tx y z x y z t
p p x y z p x y z t
pR
Cp
100000 Pa
Model Numerical Methods(Semi-implicit scheme)
Pressure gradient forceand Divergence
(Implicit Scheme)
Advection(Explicit
Scheme)Eddy
diffusion (Explicit
scheme)
Pressure gradient forceand Divergence
(Implicit Scheme)
Model
Semi-Implicit Scheme
t
x
n n n n
t x x
1 11
( )
: uncentered coefficient
15.0
Model
Terrain-following Coordinate
0
1
Model
Coordinate Transformation
x
x
x
x
z f
f
f fx
fB
,
y
y
y
y
z f
f
f fy
fB
,
z
z
B
f
fz
f .
ModelElliptic Partial differential
Equation
C
xx
n
yy
n
zz
n
f
xy
n
xz
n
fyz
n
f
x
n
y
n
z
n
f
e
xC
yC
Cx y
Cx
Cy
Cx
Cy
C
C r
2 1
2
2 1
2
2 1
2
2 1 2 1 2 1
1 1 1
For a point
Model
Coefficients
C C A mx x y y p 2 C e 1
C C B B E B mz z x x x y z p 2 2 1 2 2/ r f u v wn n n n n ( , , , , )
C Bx z x 2 C x x A R C Cv p t 2 2
C By z y y y 2 C B R C v t
C A m Bx p xf
2
x
E g B z f 1 2 2 t
C A m By p yf
2
y
0 6 5.
C A mB
BB B
BB
B EB
B E
z px
xx
f
yy
y
f
zz
fz
f
2
1 2 2
x
y
E
ProblemModel
Total=l . m . k=300,000 points
~ (300,000 x 300,000) Sparse Matrix
x: 100 grid points (l=100)y: 100 grid points (m=100)z: 30 grid points (k=30)
HopeModel
Multigrid Method
Multigrid Method
step 1
step 2
step 3
step 4
step 5
V(N1,N2) cycle
Multigrid Method
step 1
Step 1: Relax , N1 sweeps (Pre-relaxation)A U fh h h
r f A U A eh h h h h h (Residual equation)
Multigrid Method
step 2
Step 2:
Relax , N1 sweeps (Pre-relaxation)A e fh h h2 2 2
f Ihh
h h2 2 r
r f A eh h h h2 2 2 2
Multigrid Method
step 3
Step 3:
Solve (Coarse grid solution)A e fh h h4 4 4
f Ihhh h4
24 2 r
Step 4: , N2 sweeps
(Coarse grid correction)
Solve (Post-relaxation)
e e I eh hhh h2 2
42 4
A e fh h h2 2 2
Multigrid Method
step 4
Multigrid Method
step 5
Step 5: , N2 sweeps (Coarse grid correction)
Solve (Post-relaxation)
U U Ih hh
h h 22 e
A U fh h h
John C. Adams (NCAR) http://www.scd.ucar.edu/css/software/mudpack
Solve 3-D linear nonseparable elliptic partial differential equation with cross-derivative terms
Second order accuracy Finite difference operator Gauss-Seidel relaxation Gaussian Elimination (coarsest grid solution)
Multigrid SolverMultigrid Method
Full weighting restriction, multilinear interpolation
Point-by-point or line-by-line relaxation 4 color ordering V-, W-, or Full Multigrid cycling Boundary conditions: Any combination of mixed, specified, or
periodic
Multigrid SolverMultigrid Method
Flexible grid sizel qm qk q
a
b
c
1 21 2
1 2
1
2
3
| || |
max
max
x xx
in
in
in
1
1 Tolerance
x = constant, y = constant, f cons t tan
Multigrid Method
Multigrid Solver
. .. .
...
V-cycle Point-by point or line-by-line relaxationMax outer iteration : 30Boundary conditions x - specified y – specified or periodic upper - specified lower – mixed
Conditions used in our modelMultigrid Method