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

Model Formulae

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)

Model Semi-Implicit Scheme : uncentered coefficient

Model Terrain-following Coordinate

Model Coordinate Transformation

Model Elliptic Partial differential Equation For a point

Model Coefficients

Problem Model 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)

Hope Model 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) (Residual equation)

Multigrid Method step 2 Step 2: Relax, N1 sweeps (Pre-relaxation)

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

Step 4:, N2 sweeps (Coarse grid correction) Solve (Post-relaxation) Multigrid Method step 4

Multigrid Method step 5 Step 5:, N2 sweeps (Coarse grid correction) Solve (Post-relaxation)

 John C. Adams (NCAR)  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 Solver Multigrid 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 Solver Multigrid Method

 Flexible grid size  Tolerance  Multigrid Method Multigrid Solver

 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 Conditions used in our model Multigrid Method