Vertical Reconstruction

Vertical Reconstruction Methods

Within the ALE or Lagrangian Remap Method (LRM), the structure of fields within cells (or layers in the case of MOM6) are reconstructed from the resolved cell means (i.e. the model variables). The most widely used reconstructions use a piecewise polynomial representation for the reconstruction within each cell. The simplest of these is the Piecewise Constant Method (PCM) which simply uses the cell mean value as a constant value throughout the cell. The reconstructed fields may be discontinuous across cell boundaries, which is inherently the case for PCM. PCM is a first order method and considered too diffusive for ALE, although it is the implicit representation in the traditional “layered” mode. A second order reconstruction if the Piecewise Linear Method (PLM) of Van Leer, 1977 [82]. Higher order reconstructions are the Piecwise Parabloic Method (PPM) of Colella and Woodward, 1984 [17], and the Piecwise Quartic Method (PQM) of White and Adcroft, 2008 [87].

Implementation

The original implementations of vertical reconstructions are available in the src/ALE directory via modules such as plm_functions(), , ppm_functions(), , regrid_edge_values(), etc. These versions were used in OM4 , etc. These versions were used in OM4 [2] but later found to have inaccuracies with regard to round-off errors that could lead to non-monotonic behaviors. A revision of the schemes was made available after comparing and porting from Hycom and are available via modules such as mom_hybgen_remap(). A recent refactoring of reconstructions for remapping was implemented via classes derived from the . A recent refactoring of reconstructions for remapping was implemented via classes derived from the recon1d_type() (also in (also in src/ALE directory).

The following table summarizes the OM4-era and Hycom-ported methods and routines, all selected by the runtime parameter REMAPPING_SCHEME. The branch points ( select case) in the code are in mom_remapping::build_reconstructions_1d()..

REMAPPING_SCHEME	Description	Functions invoked (from MOM_remapping::build_reconstructions_1d()) )
PCM	Piecewise Constant Method	pcm_functions::pcm_reconstruction()
PLM	Monotonized Piecewise Linear Method [87]	plm_functions::plm_reconstruction() (calls (calls plm_functions::plm_slope_wa() and and plm_functions::plm_monotonized_slope()) (opt. ) (opt. plm_functions::plm_boundary_extrapolation()) )
PLM_HYBGEN	Piecewise Linear Method, ported from Hycom [17]	mom_hybgen_remap::hybgen_plm_coefs() (opt. (opt. plm_functions::plm_boundary_extrapolation()) )
PPM_H4	Piecewise Parabolic Method with explicit 4th order edge values [87]	regrid_edge_values::edge_values_explicit_h4() ppm_functions::ppm_reconstruction() (opt. (opt. ppm_functions::ppm_boundary_extrapolation()) )
PPM_IH4	Piecewise Parabolic Method with implicit 4th order edge values [87]	regrid_edge_values::edge_values_implicit_h4() ppm_functions::ppm_reconstruction() (opt. (opt. ppm_functions::ppm_boundary_extrapolation()) )
PPM_HYBGEN	Piecewise Parabolic Method with quasi-4th order edge values using PLM [17]	mom_hybgen_remap::hybgen_ppm_coefs() (opt. (opt. ppm_functions::ppm_boundary_extrapolation()) )
PPM_CW	(should be equivalent to PPM_HYBGEN)	regrid_edge_values::edge_values_explicit_h4cw() ppm_functions::ppm_monotonicity() ppm_functions::ppm_reconstruction() (calls (calls ppm_functions::ppm_limiter_standard()) (opt. ) (opt. ppm_functions::ppm_boundary_extrapolation()) )
WENO_HYBGEN	Piecewise Parabolic Method with WENO edge values, ported from Hycom	mom_hybgen_remap::hybgen_weno_coefs() ppm_functions::ppm_reconstruction() (calls (calls ppm_functions::ppm_limiter_standard()) (opt. ) (opt. ppm_functions::ppm_boundary_extrapolation()) )
PQM_IH4IH3	Piecewise Quartic Method with implicit quasi-4th order edge values and 3rd order edge slopes [87]	regrid_edge_values::edge_values_implicit_h4() regrid_edge_values::edge_slopes_implicit_h3() pqm_functions::pqm_reconstruction() (calls (calls pqm_functions::pqm_limiter()) (opt. ) (opt. pqm_functions::pqm_boundary_extrapolation_v1()) )
PQM_IH6IH5	Piecewise Quartic Method with implicit quasi-6th order edge values and 5rd order edge slopes [87]	regrid_edge_values::edge_values_implicit_h6() regrid_edge_values::edge_slopes_implicit_h5() pqm_functions::pqm_reconstruction() (calls (calls pqm_functions::pqm_limiter()) (opt. ) (opt. pqm_functions::pqm_boundary_extrapolation_v1()) )

The following table summarizes the newly refactored methods based on the class recon1d_type::recon1d(). These are also controlled by the runtime parameter . These are also controlled by the runtime parameter REMAPPING_SCHEME but the branch point is in the form of a type allocation during initialization in mom_remapping::setreconstructiontype()..

REMAPPING_SCHEME	Description	Module
C_PCM	Piecewise Constant Method (equivalent to PCM)	recon1d_pcm()
C_PLM_CW	Piecewise Linear Method (faithful to Colella and Woodward [17])	recon1d_plm_cw()
C_PLM_HYBGEN	PLM (equivalent to PLM_HYBGEN)	recon1d_plm_hybgen()
C_MPLM_WA	Monotonized Piecewise Linear Method (faithful to White and Adcroft [87])	recon1d_mplm_wa()
C_MPLM_WA_POLY	MPLM using polynomial representation (euivalent to PLM)	recon1d_mplm_wa_poly()
C_EMPLM_WA	Boundary extrapolation of MPLM_WA (faithful to White and Adcroft [87])	recon1d_emplm_wa()
C_EMPLM_WA_POLY	Boundary extrapolation of MPLM using polynomial repesentation (equivalent to PLM)	recon1d_emplm_wa_poly()
C_PLM_CWK	Piecewise Linear Method in index space (grid independent)	recon1d_plm_cwk()
C_MPLM_CWK	Monotonized Piecewise Linear Method in index space (grid independent)	recon1d_mplm_cwk()
C_EMPLM_CWK	Boundary extrapolatino of Monotonized Piecewise Linear Method in index space (grid independent)	recon1d_emplm_cwk()
C_PPM_CW	Piecewise Linear Method (faithful to Colella and Woodward [17])	recon1d_ppm_cw()
C_PPM_HYBGEN	PPM (equivalent to PPM_HYBGEN)	recon1d_ppm_hybgen()
C_PPM_H4_2018	(equivalent to PPM_H4 with answers circa 2018)	recon1d_ppm_h4_2018()
C_PPM_H4_2019	(equivalent to PPM_H4 with answers post 2019)	recon1d_ppm_h4_2019()
C_PPM_CWK	Piecewise Parabolic Method in index space (grid independent)	recon1d_ppm_cwk()
C_EPPM_CWK	Piecewise Parabolic Method in index space (grid independent)	recon1d_eppm_cwk() (extends (extends recon1d_ppm_cwk()) )

The motivation for some of the schemes in the last table was to recover certain numerical of computationsl properties, summarized in the next table.

REMAPPING_SCHEME	Representation	Globally monotonic	Consistent	Grid dependent	Uniform test
PCM	Single scalar	Yes	Yes	No	Pass
PLM	Polynomial	Forced		Yes	Fail
PLM_HYBGEN	Polynomial	No		Yes	Fail
PPM_H4	Edge values			Yes	Fail
PPM_IH4	Edge values			Yes	Fail
PPM_HYBGEN	Edge values			Yes	Fail
PPM_CW	Edge values			Yes	Fail
WENO_HYBGEN	Edge values			Yes	Fail
PQM_IH4IH3	Polynomial			Yes	Fail
PQM_IH6IH5	Polynomial			Yes	Fail
C_PCM	Single scalar	Yes	Yes	No	Pass
C_PLM_CW	Edge values	No	Yes	Yes	Pass
C_PLM_HYBGEN	Edge values	No	Yes	Yes	Pass
C_MPLM_WA	Edge values	Yes	No	Yes	Pass
C_MPLM_WA_POLY	Polynomial	Yes		Yes	Pass
C_EMPLM_WA	Edge values	Yes	No	Yes	Pass
C_EMPLM_WA_POLY	Polynomial	No		Yes	Pass
C_PLM_CWK	Edge values	Yes	Yes	No	Pass
C_MPLM_CWK	Edge values	Yes	Yes	No	Pass
C_EMPLM_CWK	Edge values	Yes	Yes	No	Pass
C_PPM_CW	Edge values	Yes	Yes	Yes	Pass
C_PPM_HYBGEN	Edge values	forced	Yes	Yes	Pass
C_PPM_H4_2018	Edge values	forced		Yes	Pass
C_PPM_H4_2019	Edge values	forced	Yes	Yes	Pass
C_PPM_CWK	Edge values	Yes	Yes	No	Pass
C_EPPM_CWK	Edge values	Yes	Yes	No	Pass

The OM4-era schemes calculate values via the function mom_remapping::average_value_ppoly() which uses reconstructions stored as the corresponding polynomial coefficients for PLM and PQM, but uses edge values for PPM. The newer class-based schemes use edge values to store the reconstructions for all schemes (except where replicating the OM4-era schemes). which uses reconstructions stored as the corresponding polynomial coefficients for PLM and PQM, but uses edge values for PPM. The newer class-based schemes use edge values to store the reconstructions for all schemes (except where replicating the OM4-era schemes).