The Application of the Multigrid Method in a

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The Application of the Multigrid Method in a Nonhydrostatic Atmospheric Model Shu-hua Chen MMM/NCAR

The Application of the Multigrid Method in a Nonhydrostatic Atmospheric Model Shu-hua Chen MMM/NCAR

Model Formulae

Model Formulae

Model Numerical Methods (Semi-implicit scheme) ¨ Pressure gradient force and Divergence (Implicit Scheme) Advection

Model Numerical Methods (Semi-implicit scheme) ¨ Pressure gradient force and Divergence (Implicit Scheme) Advection (Explicit Scheme) Eddy diffusion (Explicit scheme)

Model Semi-Implicit Scheme : uncentered coefficient

Model Semi-Implicit Scheme : uncentered coefficient

Model Terrain-following Coordinate

Model Terrain-following Coordinate

Model Coordinate Transformation

Model Coordinate Transformation

Model Elliptic Partial differential Equation For a point

Model Elliptic Partial differential Equation For a point

Model Coefficients

Model Coefficients

Model Problem x: 100 grid points (l=100) y: 100 grid points (m=100) z: 30

Model Problem x: 100 grid points (l=100) y: 100 grid points (m=100) z: 30 grid points (k=30) Total=l. m. k=300, 000 points ~ (300, 000 x 300, 000) Sparse Matrix

Model Hope Multigrid Method

Model Hope Multigrid Method

Multigrid Method V(N 1, N 2) cycle step 1 step 5 step 4 step

Multigrid Method V(N 1, N 2) cycle step 1 step 5 step 4 step 2 step 3

Multigrid Method step 1 Step 1: Relax , N 1 sweeps (Pre-relaxation) (Residual equation)

Multigrid Method step 1 Step 1: Relax , N 1 sweeps (Pre-relaxation) (Residual equation)

Multigrid Method step 2 Step 2: Relax , N 1 sweeps (Pre-relaxation)

Multigrid Method step 2 Step 2: Relax , N 1 sweeps (Pre-relaxation)

Multigrid Method Step 3: Solve (Coarse grid solution) step 3

Multigrid Method Step 3: Solve (Coarse grid solution) step 3

Multigrid Method Step 4: , N 2 sweeps (Coarse grid correction) step 4 Solve

Multigrid Method Step 4: , N 2 sweeps (Coarse grid correction) step 4 Solve (Post-relaxation)

Multigrid Method step 5 Step 5: , N 2 sweeps (Coarse grid correction) Solve

Multigrid Method step 5 Step 5: , N 2 sweeps (Coarse grid correction) Solve (Post-relaxation)

Multigrid Method Multigrid Solver · John C. Adams (NCAR) http: //www. scd. ucar. edu/css/software/mudpack

Multigrid Method Multigrid Solver · 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 Method Multigrid Solver · Full weighting restriction, multilinear interpolation · Point-by-point or line-by-line

Multigrid Method Multigrid Solver · 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 Method Multigrid Solver · Flexible grid size · · Tolerance

Multigrid Method Multigrid Solver · Flexible grid size · · Tolerance

Multigrid Method Conditions used in our model . . . . · V-cycle ·Point-by

Multigrid Method Conditions used in our model . . . . · V-cycle ·Point-by point or line-by-line relaxation ·Max outer iteration : 30 ·Boundary conditions x - specified y – specified or periodic upper - specified lower – mixed