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Numerical Algorithm Development and Testing in HYCOM

This project focuses on validating HYCOM through standardized tests and improving basic algorithms. Evaluation includes various advection schemes like MPDATA, CS-FCT, UTOPIA, WENO for applications like gravitational adjustment, density anomalies, and advection errors. Additionally, the study delves into numerical cabbeling and discontinuous Galerkin methods for enhanced computational accuracy.

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Numerical Algorithm Development and Testing in HYCOM

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  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 Viscosity=100 Hybrid mode

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

  7. SEOM 100 SCRUM 100 SEOM 50 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(Dt^2, Dx^(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 • r = r(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

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