Numerical Algorithm Development and Testing in
HYCOM
Process Oriented Test Problems
• Model validation against known solutions
• Test/improve basic algorithms
• Standardized tests
Gravitational Adjustment (advection)
Seamount (Pressure Gradient Errors)
Overflows (mixing)
Advection Schemes
• Evaluation of several advections schemes (MPDATA, CS-FCT, UTOPIA, WENO)
• Applications/Tests: Gravitational Adjustment Density Anomalies due to T/S advection
Gravitational Adjustment
HYCOM Vertical Coordinates
MICOM mode
Hybrid mode
Viscosity=100
HYCOM with z-levels
viscosity=100
Viscosity=0
WENO5-RK2
SEOM 100
SEOM 50
SCRUM 100
SCRUM 50
Advection Schemes
• Centered Difference
• MPDATA
• UTOPIA
• WENO
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
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
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
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
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
Numerical Cabbeling
• Numerical mixing may lead to cabbeling = (T,S)
• How do these schemes preserve correlations?
Density Anomalies in 1D tests
T/S Advection in Square Basin
Salinity/Density error
Spatial Discretization
• Complex Coastlines• Improve Geometry
Representation• Variable Resolution
Grids (coupled coastal-basin scale simulations)
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
Issues
• Robust advection scheme to handle layer outcropping and interaction with topography
• Experimental shallow water high-order FV model
• Extension to DGM and multilayers.
DGM Advection