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