1. Numerical Algorithm Development and Testing in HYCOM

2. Process Oriented Test Problems Model validation against known solutions Test/improve basic algorithms Standardized tests Gravitational Adjustment (advection) Seamount (Pressure Gradient Errors) Overflows (mixing)

3. Advection Schemes Evaluation of several advections schemes (MPDATA, CS-FCT, UTOPIA, WENO) Applications/Tests: Gravitational Adjustment Density Anomalies due to T/S advection

4. Gravitational Adjustment

5. HYCOM Vertical Coordinates MICOM mode Hybrid mode Viscosity=100

6. HYCOM with z-levels viscosity=100 Viscosity=0 WENO5-RK2

7. SEOM 100 SEOM 50 SCRUM 100 SCRUM 50

8. Advection Schemes Centered Difference MPDATA UTOPIA WENO

9. Centered Differences High order centered interpolation to cell edges Leap-Frog Trapezoidal time stepping O(  t^2,  x^(2n) ) No dissipation errors due to spatial discretization Gibbs oscillation controlled with FCT

10. MPDATA Based on Donor Cell scheme TE Estimate corrects dissipative DC Positivity preserving Non-oscillatory option a la FCT 2nd order in time/space, 3rd order option Not dependent on positive offset anymore

11. UTOPIA Uniform third order polynomial interpolation algorithm Upwind biased interpolation 3rd order in space/time for constant advection, otherwise only 2nd order. multidimensional Universal Limiter option Large stencil

12. WENO Weighed Essentially Non-Oscillatory Adjust stencil based on smoothness ENO criterion: Newton divided differences WENO uses a combination of these estimates to extend the order in smooth regions to O(2k-1) Expensive

13. WENO Weights depends on smoothness of solution TVD RK scheme for time integration High order extended right up to discontinuity Smooth extrema are preserved CPU intensive Large stencil

14. Numerical Cabbeling Numerical mixing may lead to cabbeling  =  (T,S) How do these schemes preserve correlations?

15. Density Anomalies in 1D tests

16. T/S Advection in Square Basin

17. Salinity/Density error

18. Spatial Discretization Complex Coastlines Improve Geometry Representation Variable Resolution Grids (coupled coastal-basin scale simulations)

19. Discontinuous Galerkin Method FEM type method Solution discontinuous at element edges Local element-wise Computations Local conservation Flux-based exchange at element edges Simple interpolation order adaptation Couple to Current Hycom

20. Issues Robust advection scheme to handle layer outcropping and interaction with topography Experimental shallow water high-order FV model Extension to DGM and multilayers.

21. DGM Advection