_Vertical_Reconstruction.dox

/*! \page Vertical_Reconstruction Vertical Reconstruction

\section section_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 \cite van_leer_1977.

Higher order reconstructions are the Piecwise Parabloic Method (PPM) of Colella and Woodward, 1984 \cite colella1984, and the Piecwise Quartic Method (PQM) of White and Adcroft, 2008 \cite white2008.

\section section_vertical_reconstruction_implementation 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 \cite Adcroft2019 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 recon1d_type (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 \cite white2008 | plm_functions::plm_reconstruction() (calls plm_functions::plm_slope_wa() and plm_functions::plm_monotonized_slope()) (opt. plm_functions::plm_boundary_extrapolation())

PLM_HYBGEN | Piecewise Linear Method, ported from Hycom \cite colella1984 | mom_hybgen_remap::hybgen_plm_coefs() (opt. plm_functions::plm_boundary_extrapolation())

PPM_H4 | Piecewise Parabolic Method with explicit 4th order edge values \cite white2008 | regrid_edge_values::edge_values_explicit_h4() ppm_functions::ppm_reconstruction() (opt. ppm_functions::ppm_boundary_extrapolation())

PPM_IH4 | Piecewise Parabolic Method with implicit 4th order edge values \cite white2008 | regrid_edge_values::edge_values_implicit_h4() ppm_functions::ppm_reconstruction() (opt. ppm_functions::ppm_boundary_extrapolation())

PPM_HYBGEN | Piecewise Parabolic Method with quasi-4th order edge values using PLM \cite colella1984 | mom_hybgen_remap::hybgen_ppm_coefs() (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 ppm_functions::ppm_limiter_standard()) (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 ppm_functions::ppm_limiter_standard()) (opt. ppm_functions::ppm_boundary_extrapolation())

PQM_IH4IH3 | Piecewise Quartic Method with implicit quasi-4th order edge values and 3rd order edge slopes \cite white2008 | regrid_edge_values::edge_values_implicit_h4() regrid_edge_values::edge_slopes_implicit_h3() pqm_functions::pqm_reconstruction() (calls pqm_functions::pqm_limiter()) (opt. pqm_functions::pqm_boundary_extrapolation_v1())

PQM_IH6IH5 | Piecewise Quartic Method with implicit quasi-6th order edge values and 5rd order edge slopes \cite white2008 | regrid_edge_values::edge_values_implicit_h6() regrid_edge_values::edge_slopes_implicit_h5() pqm_functions::pqm_reconstruction() (calls pqm_functions::pqm_limiter()) (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 `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 \cite colella1984) | 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 \cite white2008) | 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 \cite white2008) | 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 \cite colella1984) | 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 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).

*/