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