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Use Riemann Solver for Fine Scale Model

Use Riemann Solver for Fine Scale Model. Hann-Ming Henry Juang RSM workshop 2011. Contents. Introducing Riemann Solver Simple 1D for easy illustration Possible 2D In shallow water equation In nonhydrostatic system How about 3D? With splitting, 2D then 1D in vertical Discussion

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Use Riemann Solver for Fine Scale Model

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  1. Use Riemann Solver for Fine Scale Model Hann-Ming Henry Juang RSM workshop 2011

  2. Contents • Introducing Riemann Solver • Simple 1D for easy illustration • Possible 2D • In shallow water equation • In nonhydrostatic system • How about 3D? • With splitting, 2D then 1D in vertical • Discussion • advantage • Future work

  3. On dimensional shallow water equation & can be written as And we can have a matrix L and it inverse matrix L-1 for the above matrix be diagonal matrix with eigenvector as &

  4. so we can find L to be then the shallow water equation can be written as or where

  5. So the procedure to solve the 1D SWE by 1) Obtain R and C from u and h at any model grid as & 2) Use advection eq to solve next time step of R by to get 3) Obtain next time step u and h and C by & 4) Back to 2) for the next time

  6. Nonhydrostatic system For Riemann solver, we let the above be or where

  7. The initial acoustic spread

  8. After 800s with different CFL

  9. 2D nonhydrostatic tests in x-z withisotherm where

  10. 2D tests in x-z (non-forcing) – Q’(t=30s)

  11. 2D tests in x-z (non-forcing) – Q’(t=60s)

  12. SWE on spherical coordinates can be written as where let

  13. rewrite the previous SWE into Then dimensional split into

  14. Discussion • 2D system should be evaluated carefully. • 3D system can apply splitting method, do 2D first then 1D vertical, so it is no problem if 2D isno probelm. • Since it becomes an advection equation, semi-Lagrangian with splitting can be used to solve it with stable integration. • The solution becomes non isotropic while CFL is larger then 1 or 2, more to examine and resolve with additional method.

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