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Advancements in Finite Volume Advection Schemes: Numerical Methods and Models

This presentation discusses the evolution of advection schemes, focusing on the performance of OSn schemes—specifically, the advantages of OS7MP over traditional methods, including PPM and limited Prather schemes. It highlights the development of the PQM framework and the resolution of outstanding issues such as pressure gradient forces in layered models. We emphasize the importance of coherent representation across various numerical algorithms in climate models while addressing challenges like grid re-mapping and effective initialization techniques. Collaborative software development and the need for coherent profiles across model components are also examined.

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Advancements in Finite Volume Advection Schemes: Numerical Methods and Models

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  1. From ECCO2 meeting at JPL, Jan 2007 • Advection: OSn schemes (one step FV schemes) • OS7 (seventh order) • MP (monotonicity preserving) less diffusive than TVD • OS7MP better than limited Prather (SOM) • Advocated OSn FV approach over PPM FV approach • Since then have developed PQM (White & Adcroft, JCP 2008) • Inspired by OS5/7 performance w.r.t. OS3 • Analytic Finite Volume Pressure Gradient Force • Solved out-standing PGF issue with partial step topography • Paramount for layered models (Adcroft et al., OM 2008) • Paramount for general coordinates but needs revision • Now concentrating on vertical representation

  2. DST3/OS3 Fits parabola to means of all three cells O(Δx3,Δt3) accurate PPM Fits parabola to mean of local cell + cubic interpolation of edge values O(Δx3+,Δt3?) ~33 smallertruncation Advection: Variance in concepts Profile of scalar from last time level Fitted parabola Flux • SOM • Retains parabola from previous time level i.e. no fitting! • Flux of mean, slope and variance • O(Δx?, Δt?) Piecewise Parabolic Method Second-Order Moments Direct Space Time Prather, 1986

  3. Evaluation in 1D • Umlimited, PPM, OS7 clearly beat DST3 but SOM wins • When limited, OS7MP beats PPM and SOM SOM OS7MP

  4. Hybrid coordinates & Kernels Common formulation • Level models (z-coordinate) • Eulerian algorithm • FV approach in 3D • Continuous interpretation of structure • Layered models • Lagrangian algorithm • FV approach in horizontal • Vertical is contained in formulation • Piecewise constant interpretation of variables • General coordinate approach? • Post-doc: Laurent White Common software • Same numerical schemes in many models • Duplicity of effort/software • Different staggering of variables • B-grid, C-grid, … • Different choice of indexing • NE [u(i,j) is to right of T(i,j)] • SW [u(i,j) is to left of T(i,j)] • Shared software? • Programmer: NikiZadeh

  5. Re-gridding & re-mapping • Re-gridding • Re-construct global profile • Continuous • Monotonic • (not conservative) • Find position of new grid • Re-mapping • Re-construct local profiles • Discontinuous • Limited (monotonic) • Conservative • Integrate to find new cell averages Starting grid/data Fit profile Find new grid Fit profiles New cell averages

  6. Uncovering inconsistencies • Choices made on fundamental issues in one part of model have ramifications for rest of model • e.g. C-grid Coriolis discretization dictates form of KE which impacts pressure gradient discretization • Hard to reconcile definition of variables • Level models use cell average to represent continuous structure • Layer models require piecewise constant and solve physically discontinuous structure • These fundamental choices make a lot of difference • MIT numerics/formulation for z-coords • We got it right (mostly) • Are self-consistent (mostly) • But do need to re-interpret variables to generalize the vertical coordinate • i.e. re-write some terms • So far, changes have always been beneficial • Cleans up loose ends • Need consistent profiles between each of • Initialization, pressure calculation, re-gridding, re-mapping • Not horizontal transport (continuity) - we think (phew!)

  7. Interpreting model variables: initialization • MITgcm partial steps + layered models • Need same density in adjacent cells for no motion • FV reconstruction • Assign “true” average • Can reconstruct “real” profile • AFV has no pressure gradient error 28 28 28 28 27 27 27 27 26 26 26 26 25.5 25 25 25 25 25 24 24

  8. Kernels • Reps for MITgcm, HyCOM, POP, ROMS, MOM, GOLD • Library of routines • Proto-typing with horizontal advective flux • Extend with vertical advection and re-gridding/re-mapping next • Fine grained • Initially • Use “soft” conventions • Enables NE <-> SW translation • Dope vectors • Works for multiple memory layouts (for different models) • BLAS- or NAG-like • Unit tests • Validates solutions • Tests features • code.google.com • search for home-kernel • Open source • Open for debate • Open for contribution (will be) • But for now, very much under our control!  • Proof of concept • Really want to extend to parameterizations Open issues: • i,j,k alternatives • Language (!) • Adjoint? (loop bounds) • Performance • Quite a few others

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