AdvectiveFluxCalculatorBase.C

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//* This file is part of the MOOSE framework
//* https://mooseframework.inl.gov
//*
//* All rights reserved, see COPYRIGHT for full restrictions
//* https://github.com/idaholab/moose/blob/master/COPYRIGHT
//*
//* Licensed under LGPL 2.1, please see LICENSE for details
//* https://www.gnu.org/licenses/lgpl-2.1.html
 
#include "AdvectiveFluxCalculatorBase.h"
#include "Conversion.h" // for stringify
 
#include "libmesh/string_to_enum.h"
#include "libmesh/mesh_tools.h"
#include "libmesh/parallel_sync.h"
 
InputParameters
AdvectiveFluxCalculatorBase::validParams()
{
  InputParameters params = ElementUserObject::validParams();
  params.addClassDescription(
      "Base class to compute K_ij (a measure of advective flux from node i to node j) "
      "and R+ and R- (which quantify amount of antidiffusion to add) in the "
      "Kuzmin-Turek FEM-TVD multidimensional scheme");
  MooseEnum flux_limiter_type("MinMod VanLeer MC superbee None", "VanLeer");
  params.addParam<MooseEnum>("flux_limiter_type",
                             flux_limiter_type,
                             "Type of flux limiter to use.  'None' means that no antidiffusion "
                             "will be added in the Kuzmin-Turek scheme");
  params.addRangeCheckedParam<Real>(
      "allowable_MB_wastage",
      5.0,
      "allowable_MB_wastage > 0.0",
      "This object will issue a memory warning if the internal node-numbering data structure "
      "wastes more than allowable_MB_wastage megabytes.  This data structure uses sequential "
      "node-numbering which is optimized for speed rather than memory efficiency");
 
  params.addRelationshipManager("ElementPointNeighborLayers",
                                Moose::RelationshipManagerType::GEOMETRIC |
                                    Moose::RelationshipManagerType::ALGEBRAIC |
                                    Moose::RelationshipManagerType::COUPLING,
                                [](const InputParameters &, InputParameters & rm_params)
                                { rm_params.set<unsigned short>("layers") = 2; });
 
  params.set<ExecFlagEnum>("execute_on", true) = {EXEC_LINEAR};
  return params;
}
 
AdvectiveFluxCalculatorBase::AdvectiveFluxCalculatorBase(const InputParameters & parameters)
  : ElementUserObject(parameters),
    _resizing_needed(true),
    _flux_limiter_type(getParam<MooseEnum>("flux_limiter_type").getEnum<FluxLimiterTypeEnum>()),
    _kij(),
    _flux_out(),
    _dflux_out_du(),
    _dflux_out_dKjk(),
    _valence(),
    _u_nodal(),
    _u_nodal_computed_by_thread(),
    _connections(),
    _number_of_nodes(0),
    _my_pid(processor_id()),
    _nodes_to_receive(),
    _nodes_to_send(),
    _pairs_to_receive(),
    _pairs_to_send(),
    _allowable_MB_wastage(getParam<Real>("allowable_MB_wastage"))
{
  if (!_execute_enum.isValueSet(EXEC_LINEAR))
    paramError(
        "execute_on",
        "The AdvectiveFluxCalculator UserObject " + name() +
            " execute_on parameter must include, at least, 'linear'.  This is to ensure that "
            "this UserObject computes all necessary quantities just before the Kernels evaluate "
            "their Residuals");
}
 
void
AdvectiveFluxCalculatorBase::timestepSetup()
{
  // If needed, size and initialize quantities appropriately, and compute _valence
  if (_resizing_needed)
  {
    /*
     * Populate _connections for all nodes that can be seen by this processor and on relevant
     * blocks
     *
     * MULTIPROC NOTE: this must loop over local elements and 2 layers of ghosted elements.
     * The Kernel will only loop over local elements, so will only use _kij, etc, for
     * linked node-node pairs that appear in the local elements.  Nevertheless, we
     * need to build _kij, etc, for the nodes in the ghosted elements in order to simplify
     * Jacobian computations
     */
    _connections.clear();
    for (const auto & elem : _fe_problem.getNonlinearEvaluableElementRange())
      if (this->hasBlocks(elem->subdomain_id()))
        for (unsigned i = 0; i < elem->n_nodes(); ++i)
          _connections.addGlobalNode(elem->node_id(i));
    _connections.finalizeAddingGlobalNodes();
    for (const auto & elem : _fe_problem.getNonlinearEvaluableElementRange())
      if (this->hasBlocks(elem->subdomain_id()))
        for (unsigned i = 0; i < elem->n_nodes(); ++i)
          for (unsigned j = 0; j < elem->n_nodes(); ++j)
            _connections.addConnection(elem->node_id(i), elem->node_id(j));
    _connections.finalizeAddingConnections();
 
    _number_of_nodes = _connections.numNodes();
 
    Real mb_wasted = (_connections.sizeSequential() - _number_of_nodes) * 4.0 / 1048576;
    gatherMax(mb_wasted);
    if (mb_wasted > _allowable_MB_wastage)
      mooseWarning(
          "In at least one processor, the sequential node-numbering internal data structure used "
          "in " +
          name() + " is using memory inefficiently.\nThe memory wasted is " +
          Moose::stringify(mb_wasted) +
          " megabytes.\n  The node-numbering data structure has been optimized for speed at the "
          "expense of memory, but that may not be an appropriate optimization for your case, "
          "because the node numbering is not close to sequential in your case.\n");
 
    // initialize _kij
    _kij.resize(_number_of_nodes);
    for (dof_id_type sequential_i = 0; sequential_i < _number_of_nodes; ++sequential_i)
      _kij[sequential_i].assign(
          _connections.sequentialConnectionsToSequentialID(sequential_i).size(), 0.0);
 
    /*
     * Build _valence[i], which is the number of times the sequential node i is encountered when
     * looping over the entire mesh (and on relevant blocks)
     *
     * MULTIPROC NOTE: this must loop over local elements and >=1 layer of ghosted elements.
     * The Kernel will only loop over local elements, so will only use _valence for
     * linked node-node pairs that appear in the local elements.  But other processors will
     * loop over neighboring elements, so avoid multiple counting of the residual and Jacobian
     * this processor must record how many times each node-node link of its local elements
     * appears in the local elements and >=1 layer, and pass that info to the Kernel
     */
    _valence.assign(_number_of_nodes, 0);
    for (const auto & elem : _fe_problem.getNonlinearEvaluableElementRange())
      if (this->hasBlocks(elem->subdomain_id()))
        for (unsigned i = 0; i < elem->n_nodes(); ++i)
        {
          const dof_id_type node_i = elem->node_id(i);
          const dof_id_type sequential_i = _connections.sequentialID(node_i);
          _valence[sequential_i] += 1;
        }
 
    _u_nodal.assign(_number_of_nodes, 0.0);
    _u_nodal_computed_by_thread.assign(_number_of_nodes, false);
    _flux_out.assign(_number_of_nodes, 0.0);
    _dflux_out_du.assign(_number_of_nodes, std::map<dof_id_type, Real>());
    for (dof_id_type sequential_i = 0; sequential_i < _number_of_nodes; ++sequential_i)
      zeroedConnection(_dflux_out_du[sequential_i], _connections.globalID(sequential_i));
    _dflux_out_dKjk.resize(_number_of_nodes);
    for (dof_id_type sequential_i = 0; sequential_i < _number_of_nodes; ++sequential_i)
    {
      const std::vector<dof_id_type> con_i =
          _connections.sequentialConnectionsToSequentialID(sequential_i);
      const std::size_t num_con_i = con_i.size();
      _dflux_out_dKjk[sequential_i].resize(num_con_i);
      for (std::size_t j = 0; j < con_i.size(); ++j)
      {
        const dof_id_type sequential_j = con_i[j];
        const std::size_t num_con_j =
            _connections.sequentialConnectionsToSequentialID(sequential_j).size();
        _dflux_out_dKjk[sequential_i][j].assign(num_con_j, 0.0);
      }
    }
 
    if (_app.n_processors() > 1)
      buildCommLists();
 
    _resizing_needed = false;
 
    // Clear and size all member vectors
    _dij.assign(_number_of_nodes, {});
    _dDij_dKij.assign(_number_of_nodes, {});
    _dDij_dKji.assign(_number_of_nodes, {});
    _dDii_dKij.assign(_number_of_nodes, {});
    _dDii_dKji.assign(_number_of_nodes, {});
    _lij.assign(_number_of_nodes, {});
    _rP.assign(_number_of_nodes, 0.0);
    _rM.assign(_number_of_nodes, 0.0);
    _drP.assign(_number_of_nodes, {});
    _drM.assign(_number_of_nodes, {});
    _drP_dk.assign(_number_of_nodes, {});
    _drM_dk.assign(_number_of_nodes, {});
    _fa.assign(_number_of_nodes, {});
    _dfa.assign(_number_of_nodes, {});
    _dFij_dKik.assign(_number_of_nodes, {});
    _dFij_dKjk.assign(_number_of_nodes, {});
  }
}
 
void
AdvectiveFluxCalculatorBase::meshChanged()
{
  ElementUserObject::meshChanged();
 
  // Signal that _kij, _valence, etc need to be rebuilt
  _resizing_needed = true;
}
 
void
AdvectiveFluxCalculatorBase::initialize()
{
  // Zero _kij and falsify _u_nodal_computed_by_thread ready for building in execute() and
  // finalize()
  _u_nodal_computed_by_thread.assign(_number_of_nodes, false);
  for (dof_id_type sequential_i = 0; sequential_i < _number_of_nodes; ++sequential_i)
    _kij[sequential_i].assign(_connections.sequentialConnectionsToSequentialID(sequential_i).size(),
                              0.0);
}
 
void
AdvectiveFluxCalculatorBase::execute()
{
  // compute _kij contributions from this element that is local to this processor
  // and record _u_nodal
  for (unsigned i = 0; i < _current_elem->n_nodes(); ++i)
  {
    const dof_id_type node_i = _current_elem->node_id(i);
    const dof_id_type sequential_i = _connections.sequentialID(node_i);
    if (!_u_nodal_computed_by_thread[sequential_i])
    {
      _u_nodal[sequential_i] = computeU(i);
      _u_nodal_computed_by_thread[sequential_i] = true;
    }
    for (unsigned j = 0; j < _current_elem->n_nodes(); ++j)
    {
      const dof_id_type node_j = _current_elem->node_id(j);
      for (unsigned qp = 0; qp < _qrule->n_points(); ++qp)
        executeOnElement(node_i, node_j, i, j, qp);
    }
  }
}
 
void
AdvectiveFluxCalculatorBase::executeOnElement(
    dof_id_type global_i, dof_id_type global_j, unsigned local_i, unsigned local_j, unsigned qp)
{
  // KT Eqn (20)
  const dof_id_type sequential_i = _connections.sequentialID(global_i);
  const unsigned index_i_to_j = _connections.indexOfGlobalConnection(global_i, global_j);
  _kij[sequential_i][index_i_to_j] += _JxW[qp] * _coord[qp] * computeVelocity(local_i, local_j, qp);
}
 
void
AdvectiveFluxCalculatorBase::threadJoin(const UserObject & uo)
{
  const auto & afc = static_cast<const AdvectiveFluxCalculatorBase &>(uo);
  // add the values of _kij computed by different threads
  for (dof_id_type sequential_i = 0; sequential_i < _number_of_nodes; ++sequential_i)
  {
    const std::size_t num_con_i =
        _connections.sequentialConnectionsToSequentialID(sequential_i).size();
    for (std::size_t j = 0; j < num_con_i; ++j)
      _kij[sequential_i][j] += afc._kij[sequential_i][j];
  }
 
  // gather the values of _u_nodal computed by different threads
  for (dof_id_type sequential_i = 0; sequential_i < _number_of_nodes; ++sequential_i)
    if (!_u_nodal_computed_by_thread[sequential_i] && afc._u_nodal_computed_by_thread[sequential_i])
      _u_nodal[sequential_i] = afc._u_nodal[sequential_i];
}
 
void
AdvectiveFluxCalculatorBase::finalize()
{
  // Overall: ensure _kij is fully built, then compute Kuzmin-Turek D, L, f^a and
  // relevant Jacobian information, and then the relevant quantities into _flux_out and
  // _dflux_out_du, _dflux_out_dKjk
 
  if (_app.n_processors() > 1)
    exchangeGhostedInfo();
 
  // Calculate KuzminTurek D matrix
  // See Eqn (32)
  // This adds artificial diffusion, which eliminates any spurious oscillations
  // The idea is that D will remove all negative off-diagonal elements when it is added to K
  // This is identical to full upwinding
  for (dof_id_type sequential_i = 0; sequential_i < _number_of_nodes; ++sequential_i)
  {
    const std::vector<dof_id_type> & con_i =
        _connections.sequentialConnectionsToSequentialID(sequential_i);
    const std::size_t num_con_i = con_i.size();
    _dij[sequential_i].assign(num_con_i, 0.0);
    _dDij_dKij[sequential_i].assign(num_con_i, 0.0);
    _dDij_dKji[sequential_i].assign(num_con_i, 0.0);
    _dDii_dKij[sequential_i].assign(num_con_i, 0.0);
    _dDii_dKji[sequential_i].assign(num_con_i, 0.0);
    const unsigned index_i_to_i =
        _connections.indexOfSequentialConnection(sequential_i, sequential_i);
    for (std::size_t j = 0; j < num_con_i; ++j)
    {
      const dof_id_type sequential_j = con_i[j];
      if (sequential_i == sequential_j)
        continue;
      const unsigned index_j_to_i =
          _connections.indexOfSequentialConnection(sequential_j, sequential_i);
      const Real kij = _kij[sequential_i][j];
      const Real kji = _kij[sequential_j][index_j_to_i];
      if ((kij <= kji) && (kij < 0.0))
      {
        _dij[sequential_i][j] = -kij;
        _dDij_dKij[sequential_i][j] = -1.0;
        _dDii_dKij[sequential_i][j] += 1.0;
      }
      else if ((kji <= kij) && (kji < 0.0))
      {
        _dij[sequential_i][j] = -kji;
        _dDij_dKji[sequential_i][j] = -1.0;
        _dDii_dKji[sequential_i][j] += 1.0;
      }
      else
        _dij[sequential_i][j] = 0.0;
      _dij[sequential_i][index_i_to_i] -= _dij[sequential_i][j];
    }
  }
 
  // Calculate KuzminTurek L matrix
  // See Fig 2: L = K + D
  for (dof_id_type sequential_i = 0; sequential_i < _number_of_nodes; ++sequential_i)
  {
    const std::vector<dof_id_type> & con_i =
        _connections.sequentialConnectionsToSequentialID(sequential_i);
    const std::size_t num_con_i = con_i.size();
    _lij[sequential_i].assign(num_con_i, 0.0);
    for (std::size_t j = 0; j < num_con_i; ++j)
      _lij[sequential_i][j] = _kij[sequential_i][j] + _dij[sequential_i][j];
  }
 
  // Compute KuzminTurek R matrices
  // See Eqns (49) and (12)
  for (dof_id_type sequential_i = 0; sequential_i < _number_of_nodes; ++sequential_i)
  {
    _rP[sequential_i] = rPlus(sequential_i, _drP[sequential_i], _drP_dk[sequential_i]);
    _rM[sequential_i] = rMinus(sequential_i, _drM[sequential_i], _drM_dk[sequential_i]);
  }
 
  // Calculate KuzminTurek f^{a} matrix
  // This is the antidiffusive flux
  // See Eqn (50)
  for (dof_id_type sequential_i = 0; sequential_i < _number_of_nodes; ++sequential_i)
  {
    const std::vector<dof_id_type> & con_i =
        _connections.globalConnectionsToSequentialID(sequential_i);
    const unsigned num_con_i = con_i.size();
    _fa[sequential_i].assign(num_con_i, 0.0);
    _dfa[sequential_i].resize(num_con_i);
    _dFij_dKik[sequential_i].resize(num_con_i);
    _dFij_dKjk[sequential_i].resize(num_con_i);
    for (std::size_t j = 0; j < num_con_i; ++j)
    {
      for (const auto & global_k : con_i)
        _dfa[sequential_i][j][global_k] = 0;
      const dof_id_type global_j = con_i[j];
      const std::vector<dof_id_type> & con_j = _connections.globalConnectionsToGlobalID(global_j);
      const unsigned num_con_j = con_j.size();
      for (const auto & global_k : con_j)
        _dfa[sequential_i][j][global_k] = 0;
      _dFij_dKik[sequential_i][j].assign(num_con_i, 0.0);
      _dFij_dKjk[sequential_i][j].assign(num_con_j, 0.0);
    }
  }
 
  for (dof_id_type sequential_i = 0; sequential_i < _number_of_nodes; ++sequential_i)
  {
    const dof_id_type global_i = _connections.globalID(sequential_i);
    const Real u_i = _u_nodal[sequential_i];
    const std::vector<dof_id_type> & con_i =
        _connections.sequentialConnectionsToSequentialID(sequential_i);
    const std::size_t num_con_i = con_i.size();
    for (std::size_t j = 0; j < num_con_i; ++j)
    {
      const dof_id_type sequential_j = con_i[j];
      if (sequential_i == sequential_j)
        continue;
      const unsigned index_j_to_i =
          _connections.indexOfSequentialConnection(sequential_j, sequential_i);
      const Real Lij = _lij[sequential_i][j];
      const Real Lji = _lij[sequential_j][index_j_to_i];
      if (Lji >= Lij) // node i is upwind of node j.
      {
        const Real Dij = _dij[sequential_i][j];
        const dof_id_type global_j = _connections.globalID(sequential_j);
        const Real u_j = _u_nodal[sequential_j];
        Real prefactor = 0.0;
        std::vector<Real> dprefactor_du(num_con_i,
                                        0.0); // dprefactor_du[j] = d(prefactor)/du[sequential_j]
        std::vector<Real> dprefactor_dKij(
            num_con_i, 0.0);      // dprefactor_dKij[j] = d(prefactor)/dKij[sequential_i][j].
        Real dprefactor_dKji = 0; // dprefactor_dKji = d(prefactor)/dKij[sequential_j][index_j_to_i]
        if (u_i >= u_j)
        {
          if (Lji <= _rP[sequential_i] * Dij)
          {
            prefactor = Lji;
            dprefactor_dKij[j] += _dDij_dKji[sequential_j][index_j_to_i];
            dprefactor_dKji += 1.0 + _dDij_dKij[sequential_j][index_j_to_i];
          }
          else
          {
            prefactor = _rP[sequential_i] * Dij;
            for (std::size_t ind_j = 0; ind_j < num_con_i; ++ind_j)
              dprefactor_du[ind_j] = _drP[sequential_i][ind_j] * Dij;
            dprefactor_dKij[j] += _rP[sequential_i] * _dDij_dKij[sequential_i][j];
            dprefactor_dKji += _rP[sequential_i] * _dDij_dKji[sequential_i][j];
            for (std::size_t ind_j = 0; ind_j < num_con_i; ++ind_j)
              dprefactor_dKij[ind_j] += _drP_dk[sequential_i][ind_j] * Dij;
          }
        }
        else
        {
          if (Lji <= _rM[sequential_i] * Dij)
          {
            prefactor = Lji;
            dprefactor_dKij[j] += _dDij_dKji[sequential_j][index_j_to_i];
            dprefactor_dKji += 1.0 + _dDij_dKij[sequential_j][index_j_to_i];
          }
          else
          {
            prefactor = _rM[sequential_i] * Dij;
            for (std::size_t ind_j = 0; ind_j < num_con_i; ++ind_j)
              dprefactor_du[ind_j] = _drM[sequential_i][ind_j] * Dij;
            dprefactor_dKij[j] += _rM[sequential_i] * _dDij_dKij[sequential_i][j];
            dprefactor_dKji += _rM[sequential_i] * _dDij_dKji[sequential_i][j];
            for (std::size_t ind_j = 0; ind_j < num_con_i; ++ind_j)
              dprefactor_dKij[ind_j] += _drM_dk[sequential_i][ind_j] * Dij;
          }
        }
        _fa[sequential_i][j] = prefactor * (u_i - u_j);
        _dfa[sequential_i][j][global_i] = prefactor;
        _dfa[sequential_i][j][global_j] = -prefactor;
        for (std::size_t ind_j = 0; ind_j < num_con_i; ++ind_j)
        {
          _dfa[sequential_i][j][_connections.globalID(con_i[ind_j])] +=
              dprefactor_du[ind_j] * (u_i - u_j);
          _dFij_dKik[sequential_i][j][ind_j] += dprefactor_dKij[ind_j] * (u_i - u_j);
        }
        _dFij_dKjk[sequential_i][j][index_j_to_i] += dprefactor_dKji * (u_i - u_j);
      }
    }
  }
 
  for (dof_id_type sequential_i = 0; sequential_i < _number_of_nodes; ++sequential_i)
  {
    const std::vector<dof_id_type> & con_i =
        _connections.sequentialConnectionsToSequentialID(sequential_i);
    const std::size_t num_con_i = con_i.size();
    for (std::size_t j = 0; j < num_con_i; ++j)
    {
      const dof_id_type sequential_j = con_i[j];
      if (sequential_i == sequential_j)
        continue;
      const unsigned index_j_to_i =
          _connections.indexOfSequentialConnection(sequential_j, sequential_i);
      if (_lij[sequential_j][index_j_to_i] < _lij[sequential_i][j]) // node_i is downwind of node_j.
      {
        _fa[sequential_i][j] = -_fa[sequential_j][index_j_to_i];
        for (const auto & dof_deriv : _dfa[sequential_j][index_j_to_i])
          _dfa[sequential_i][j][dof_deriv.first] = -dof_deriv.second;
        for (std::size_t k = 0; k < num_con_i; ++k)
          _dFij_dKik[sequential_i][j][k] = -_dFij_dKjk[sequential_j][index_j_to_i][k];
        const std::size_t num_con_j =
            _connections.sequentialConnectionsToSequentialID(sequential_j).size();
        for (std::size_t k = 0; k < num_con_j; ++k)
          _dFij_dKjk[sequential_i][j][k] = -_dFij_dKik[sequential_j][index_j_to_i][k];
      }
    }
  }
 
  // zero _flux_out and its derivatives
  _flux_out.assign(_number_of_nodes, 0.0);
  // The derivatives are a bit complicated.
  // If i is upwind of a node "j" then _flux_out[i] depends on all nodes connected to i.
  // But if i is downwind of a node "j" then _flux_out depends on all nodes connected with node
  // j.
  _dflux_out_du.assign(_number_of_nodes, std::map<dof_id_type, Real>());
  for (dof_id_type sequential_i = 0; sequential_i < _number_of_nodes; ++sequential_i)
    for (const auto & j : _connections.globalConnectionsToSequentialID(sequential_i))
    {
      _dflux_out_du[sequential_i][j] = 0.0;
      for (const auto & neighbors_j : _connections.globalConnectionsToGlobalID(j))
        _dflux_out_du[sequential_i][neighbors_j] = 0.0;
    }
  _dflux_out_dKjk.resize(_number_of_nodes);
  for (dof_id_type sequential_i = 0; sequential_i < _number_of_nodes; ++sequential_i)
  {
    const std::vector<dof_id_type> & con_i =
        _connections.sequentialConnectionsToSequentialID(sequential_i);
    const std::size_t num_con_i = con_i.size();
    _dflux_out_dKjk[sequential_i].resize(num_con_i);
    for (std::size_t j = 0; j < num_con_i; ++j)
    {
      const dof_id_type sequential_j = con_i[j];
      const std::vector<dof_id_type> & con_j =
          _connections.sequentialConnectionsToSequentialID(sequential_j);
      _dflux_out_dKjk[sequential_i][j].assign(con_j.size(), 0.0);
    }
  }
 
  // Add everything together
  // See step 3 in Fig 2, noting Eqn (36)
  for (dof_id_type sequential_i = 0; sequential_i < _number_of_nodes; ++sequential_i)
  {
    const std::vector<dof_id_type> & con_i =
        _connections.sequentialConnectionsToSequentialID(sequential_i);
    const size_t num_con_i = con_i.size();
    const dof_id_type index_i_to_i =
        _connections.indexOfSequentialConnection(sequential_i, sequential_i);
    for (std::size_t j = 0; j < num_con_i; ++j)
    {
      const dof_id_type sequential_j = con_i[j];
      const dof_id_type global_j = _connections.globalID(sequential_j);
      const Real u_j = _u_nodal[sequential_j];
 
      // negative sign because residual = -Lu (KT equation (19))
      _flux_out[sequential_i] -= _lij[sequential_i][j] * u_j + _fa[sequential_i][j];
 
      _dflux_out_du[sequential_i][global_j] -= _lij[sequential_i][j];
      for (const auto & dof_deriv : _dfa[sequential_i][j])
        _dflux_out_du[sequential_i][dof_deriv.first] -= dof_deriv.second;
 
      _dflux_out_dKjk[sequential_i][index_i_to_i][j] -= 1.0 * u_j; // from the K in L = K + D
 
      if (sequential_j == sequential_i)
        for (dof_id_type k = 0; k < num_con_i; ++k)
          _dflux_out_dKjk[sequential_i][index_i_to_i][k] -= _dDii_dKij[sequential_i][k] * u_j;
      else
        _dflux_out_dKjk[sequential_i][index_i_to_i][j] -= _dDij_dKij[sequential_i][j] * u_j;
      for (dof_id_type k = 0; k < num_con_i; ++k)
        _dflux_out_dKjk[sequential_i][index_i_to_i][k] -= _dFij_dKik[sequential_i][j][k];
 
      if (sequential_j == sequential_i)
        for (unsigned k = 0; k < con_i.size(); ++k)
        {
          const unsigned index_k_to_i =
              _connections.indexOfSequentialConnection(con_i[k], sequential_i);
          _dflux_out_dKjk[sequential_i][k][index_k_to_i] -= _dDii_dKji[sequential_i][k] * u_j;
        }
      else
      {
        const unsigned index_j_to_i =
            _connections.indexOfSequentialConnection(sequential_j, sequential_i);
        _dflux_out_dKjk[sequential_i][j][index_j_to_i] -= _dDij_dKji[sequential_i][j] * u_j;
      }
      for (unsigned k = 0;
           k < _connections.sequentialConnectionsToSequentialID(sequential_j).size();
           ++k)
        _dflux_out_dKjk[sequential_i][j][k] -= _dFij_dKjk[sequential_i][j][k];
    }
  }
}
 
Real
AdvectiveFluxCalculatorBase::rPlus(dof_id_type sequential_i,
                                   std::vector<Real> & dlimited_du,
                                   std::vector<Real> & dlimited_dk) const
{
  const std::size_t num_con = _connections.sequentialConnectionsToSequentialID(sequential_i).size();
  dlimited_du.assign(num_con, 0.0);
  dlimited_dk.assign(num_con, 0.0);
  if (_flux_limiter_type == FluxLimiterTypeEnum::None)
    return 0.0;
  std::vector<Real> dp_du;
  std::vector<Real> dp_dk;
  const Real p = PQPlusMinus(sequential_i, PQPlusMinusEnum::PPlus, dp_du, dp_dk);
  if (p == 0.0)
    // Comment after Eqn (49): if P=0 then there's no antidiffusion, so no need to remove it
    return 1.0;
  std::vector<Real> dq_du;
  std::vector<Real> dq_dk;
  const Real q = PQPlusMinus(sequential_i, PQPlusMinusEnum::QPlus, dq_du, dq_dk);
 
  const Real r = q / p;
  Real limited;
  Real dlimited_dr;
  limitFlux(1.0, r, limited, dlimited_dr);
 
  const Real p2 = std::pow(p, 2);
  for (std::size_t j = 0; j < num_con; ++j)
  {
    const Real dr_du = dq_du[j] / p - q * dp_du[j] / p2;
    const Real dr_dk = dq_dk[j] / p - q * dp_dk[j] / p2;
    dlimited_du[j] = dlimited_dr * dr_du;
    dlimited_dk[j] = dlimited_dr * dr_dk;
  }
  return limited;
}
 
Real
AdvectiveFluxCalculatorBase::rMinus(dof_id_type sequential_i,
                                    std::vector<Real> & dlimited_du,
                                    std::vector<Real> & dlimited_dk) const
{
  const std::size_t num_con = _connections.sequentialConnectionsToSequentialID(sequential_i).size();
  dlimited_du.assign(num_con, 0.0);
  dlimited_dk.assign(num_con, 0.0);
  if (_flux_limiter_type == FluxLimiterTypeEnum::None)
    return 0.0;
  std::vector<Real> dp_du;
  std::vector<Real> dp_dk;
  const Real p = PQPlusMinus(sequential_i, PQPlusMinusEnum::PMinus, dp_du, dp_dk);
  if (p == 0.0)
    // Comment after Eqn (49): if P=0 then there's no antidiffusion, so no need to remove it
    return 1.0;
  std::vector<Real> dq_du;
  std::vector<Real> dq_dk;
  const Real q = PQPlusMinus(sequential_i, PQPlusMinusEnum::QMinus, dq_du, dq_dk);
 
  const Real r = q / p;
  Real limited;
  Real dlimited_dr;
  limitFlux(1.0, r, limited, dlimited_dr);
 
  const Real p2 = std::pow(p, 2);
  for (std::size_t j = 0; j < num_con; ++j)
  {
    const Real dr_du = dq_du[j] / p - q * dp_du[j] / p2;
    const Real dr_dk = dq_dk[j] / p - q * dp_dk[j] / p2;
    dlimited_du[j] = dlimited_dr * dr_du;
    dlimited_dk[j] = dlimited_dr * dr_dk;
  }
  return limited;
}
 
void
AdvectiveFluxCalculatorBase::limitFlux(Real a, Real b, Real & limited, Real & dlimited_db) const
{
  limited = 0.0;
  dlimited_db = 0.0;
  if (_flux_limiter_type == FluxLimiterTypeEnum::None)
    return;
 
  if ((a >= 0.0 && b <= 0.0) || (a <= 0.0 && b >= 0.0))
    return;
  const Real s = (a > 0.0 ? 1.0 : -1.0);
 
  const Real lal = std::abs(a);
  const Real lbl = std::abs(b);
  const Real dlbl = (b >= 0.0 ? 1.0 : -1.0); // d(lbl)/db
  switch (_flux_limiter_type)
  {
    case FluxLimiterTypeEnum::MinMod:
    {
      if (lal <= lbl)
      {
        limited = s * lal;
        dlimited_db = 0.0;
      }
      else
      {
        limited = s * lbl;
        dlimited_db = s * dlbl;
      }
      return;
    }
    case FluxLimiterTypeEnum::VanLeer:
    {
      limited = s * 2 * lal * lbl / (lal + lbl);
      dlimited_db = s * 2 * lal * (dlbl / (lal + lbl) - lbl * dlbl / std::pow(lal + lbl, 2));
      return;
    }
    case FluxLimiterTypeEnum::MC:
    {
      const Real av = 0.5 * std::abs(a + b);
      if (2 * lal <= av && lal <= lbl)
      {
        // 2 * lal is the smallest
        limited = s * 2.0 * lal;
        dlimited_db = 0.0;
      }
      else if (2 * lbl <= av && lbl <= lal)
      {
        // 2 * lbl is the smallest
        limited = s * 2.0 * lbl;
        dlimited_db = s * 2.0 * dlbl;
      }
      else
      {
        // av is the smallest
        limited = s * av;
        // if (a>0 and b>0) then d(av)/db = 0.5 = 0.5 * dlbl
        // if (a<0 and b<0) then d(av)/db = -0.5 = 0.5 * dlbl
        // if a and b have different sign then limited=0, above
        dlimited_db = s * 0.5 * dlbl;
      }
      return;
    }
    case FluxLimiterTypeEnum::superbee:
    {
      const Real term1 = std::min(2.0 * lal, lbl);
      const Real term2 = std::min(lal, 2.0 * lbl);
      if (term1 >= term2)
      {
        if (2.0 * lal <= lbl)
        {
          limited = s * 2 * lal;
          dlimited_db = 0.0;
        }
        else
        {
          limited = s * lbl;
          dlimited_db = s * dlbl;
        }
      }
      else
      {
        if (lal <= 2.0 * lbl)
        {
          limited = s * lal;
          dlimited_db = 0.0;
        }
        else
        {
          limited = s * 2.0 * lbl;
          dlimited_db = s * 2.0 * dlbl;
        }
      }
      return;
    }
    default:
      return;
  }
}
 
const std::map<dof_id_type, Real> &
AdvectiveFluxCalculatorBase::getdFluxOutdu(dof_id_type node_i) const
{
  return _dflux_out_du[_connections.sequentialID(node_i)];
}
 
const std::vector<std::vector<Real>> &
AdvectiveFluxCalculatorBase::getdFluxOutdKjk(dof_id_type node_i) const
{
  return _dflux_out_dKjk[_connections.sequentialID(node_i)];
}
 
Real
AdvectiveFluxCalculatorBase::getFluxOut(dof_id_type node_i) const
{
  return _flux_out[_connections.sequentialID(node_i)];
}
 
unsigned
AdvectiveFluxCalculatorBase::getValence(dof_id_type node_i) const
{
  return _valence[_connections.sequentialID(node_i)];
}
 
void
AdvectiveFluxCalculatorBase::zeroedConnection(std::map<dof_id_type, Real> & the_map,
                                              dof_id_type node_i) const
{
  the_map.clear();
  for (const auto & node_j : _connections.globalConnectionsToGlobalID(node_i))
    the_map[node_j] = 0.0;
}
 
Real
AdvectiveFluxCalculatorBase::PQPlusMinus(dof_id_type sequential_i,
                                         const PQPlusMinusEnum pq_plus_minus,
                                         std::vector<Real> & derivs,
                                         std::vector<Real> & dpqdk) const
{
  // Find the value of u at sequential_i
  const Real u_i = _u_nodal[sequential_i];
 
  // Connections to sequential_i
  const std::vector<dof_id_type> con_i =
      _connections.sequentialConnectionsToSequentialID(sequential_i);
  const std::size_t num_con = con_i.size();
  // The neighbor number of sequential_i to sequential_i
  const unsigned i_index_i = _connections.indexOfSequentialConnection(sequential_i, sequential_i);
 
  // Initialize the results
  Real result = 0.0;
  derivs.assign(num_con, 0.0);
  dpqdk.assign(num_con, 0.0);
 
  // Sum over all nodes connected with node_i.
  for (std::size_t j = 0; j < num_con; ++j)
  {
    const dof_id_type sequential_j = con_i[j];
    if (sequential_j == sequential_i)
      continue;
    const Real kentry = _kij[sequential_i][j];
 
    // Find the value of u at node_j
    const Real u_j = _u_nodal[sequential_j];
    const Real ujminusi = u_j - u_i;
 
    // Evaluate the i-j contribution to the result
    switch (pq_plus_minus)
    {
      case PQPlusMinusEnum::PPlus:
      {
        if (ujminusi < 0.0 && kentry < 0.0)
        {
          result += kentry * ujminusi;
          derivs[j] += kentry;
          derivs[i_index_i] -= kentry;
          dpqdk[j] += ujminusi;
        }
        break;
      }
      case PQPlusMinusEnum::PMinus:
      {
        if (ujminusi > 0.0 && kentry < 0.0)
        {
          result += kentry * ujminusi;
          derivs[j] += kentry;
          derivs[i_index_i] -= kentry;
          dpqdk[j] += ujminusi;
        }
        break;
      }
      case PQPlusMinusEnum::QPlus:
      {
        if (ujminusi > 0.0 && kentry > 0.0)
        {
          result += kentry * ujminusi;
          derivs[j] += kentry;
          derivs[i_index_i] -= kentry;
          dpqdk[j] += ujminusi;
        }
        break;
      }
      case PQPlusMinusEnum::QMinus:
      {
        if (ujminusi < 0.0 && kentry > 0.0)
        {
          result += kentry * ujminusi;
          derivs[j] += kentry;
          derivs[i_index_i] -= kentry;
          dpqdk[j] += ujminusi;
        }
        break;
      }
    }
  }
 
  return result;
}
 
void
AdvectiveFluxCalculatorBase::buildCommLists()
{
  _nodes_to_receive.clear();
  for (const auto & elem : _fe_problem.getNonlinearEvaluableElementRange())
    if (this->hasBlocks(elem->subdomain_id()))
    {
      const processor_id_type elem_pid = elem->processor_id();
      if (elem_pid != _my_pid)
      {
        if (_nodes_to_receive.find(elem_pid) == _nodes_to_receive.end())
          _nodes_to_receive[elem_pid] = std::vector<dof_id_type>();
        for (unsigned i = 0; i < elem->n_nodes(); ++i)
          if (std::find(_nodes_to_receive[elem_pid].begin(),
                        _nodes_to_receive[elem_pid].end(),
                        elem->node_id(i)) == _nodes_to_receive[elem_pid].end())
            _nodes_to_receive[elem_pid].push_back(elem->node_id(i));
      }
    }
 
  // exchange this info with other processors, building _nodes_to_send at the same time
  _nodes_to_send.clear();
  auto nodes_action_functor = [this](processor_id_type pid, const std::vector<dof_id_type> & nts)
  { _nodes_to_send[pid] = nts; };
  Parallel::push_parallel_vector_data(this->comm(), _nodes_to_receive, nodes_action_functor);
 
  // At the moment,  _nodes_to_send and _nodes_to_receive contain global node numbers
  // It is slightly more efficient to convert to sequential node numbers
  // so that we don't have to keep doing things like
  // _u_nodal[_connections.sequentialID(_nodes_to_send[pid][i])
  // every time we send/receive
  for (auto & kv : _nodes_to_send)
  {
    const processor_id_type pid = kv.first;
    const std::size_t num_nodes = kv.second.size();
    for (unsigned i = 0; i < num_nodes; ++i)
      _nodes_to_send[pid][i] = _connections.sequentialID(_nodes_to_send[pid][i]);
  }
  for (auto & kv : _nodes_to_receive)
  {
    const processor_id_type pid = kv.first;
    const std::size_t num_nodes = kv.second.size();
    for (unsigned i = 0; i < num_nodes; ++i)
      _nodes_to_receive[pid][i] = _connections.sequentialID(_nodes_to_receive[pid][i]);
  }
 
  // Build pairs_to_receive
  _pairs_to_receive.clear();
  for (const auto & elem : _fe_problem.getNonlinearEvaluableElementRange())
    if (this->hasBlocks(elem->subdomain_id()))
    {
      const processor_id_type elem_pid = elem->processor_id();
      if (elem_pid != _my_pid)
      {
        if (_pairs_to_receive.find(elem_pid) == _pairs_to_receive.end())
          _pairs_to_receive[elem_pid] = std::vector<std::pair<dof_id_type, dof_id_type>>();
        for (unsigned i = 0; i < elem->n_nodes(); ++i)
          for (unsigned j = 0; j < elem->n_nodes(); ++j)
          {
            std::pair<dof_id_type, dof_id_type> the_pair(elem->node_id(i), elem->node_id(j));
            if (std::find(_pairs_to_receive[elem_pid].begin(),
                          _pairs_to_receive[elem_pid].end(),
                          the_pair) == _pairs_to_receive[elem_pid].end())
              _pairs_to_receive[elem_pid].push_back(the_pair);
          }
      }
    }
 
  _pairs_to_send.clear();
  auto pairs_action_functor =
      [this](processor_id_type pid, const std::vector<std::pair<dof_id_type, dof_id_type>> & pts)
  { _pairs_to_send[pid] = pts; };
  Parallel::push_parallel_vector_data(this->comm(), _pairs_to_receive, pairs_action_functor);
 
  // _pairs_to_send and _pairs_to_receive have been built using global node IDs
  // since all processors know about that.  However, using global IDs means
  // that every time we send/receive, we keep having to do things like
  // _kij[_connections.sequentialID(_pairs_to_send[pid][i].first)][_connections.indexOfGlobalConnection(_pairs_to_send[pid][i].first,
  // _pairs_to_send[pid][i].second)] which is quite inefficient.  So:
  for (auto & kv : _pairs_to_send)
  {
    const processor_id_type pid = kv.first;
    const std::size_t num_pairs = kv.second.size();
    for (unsigned i = 0; i < num_pairs; ++i)
    {
      _pairs_to_send[pid][i].second = _connections.indexOfGlobalConnection(
          _pairs_to_send[pid][i].first, _pairs_to_send[pid][i].second);
      _pairs_to_send[pid][i].first = _connections.sequentialID(_pairs_to_send[pid][i].first);
    }
  }
  for (auto & kv : _pairs_to_receive)
  {
    const processor_id_type pid = kv.first;
    const std::size_t num_pairs = kv.second.size();
    for (unsigned i = 0; i < num_pairs; ++i)
    {
      _pairs_to_receive[pid][i].second = _connections.indexOfGlobalConnection(
          _pairs_to_receive[pid][i].first, _pairs_to_receive[pid][i].second);
      _pairs_to_receive[pid][i].first = _connections.sequentialID(_pairs_to_receive[pid][i].first);
    }
  }
}
 
void
AdvectiveFluxCalculatorBase::exchangeGhostedInfo()
{
  // Exchange ghosted _u_nodal information with other processors
  std::map<processor_id_type, std::vector<Real>> unodal_to_send;
  for (const auto & kv : _nodes_to_send)
  {
    const processor_id_type pid = kv.first;
    unodal_to_send[pid] = std::vector<Real>();
    for (const auto & nd : kv.second)
      unodal_to_send[pid].push_back(_u_nodal[nd]);
  }
 
  auto unodal_action_functor =
      [this](processor_id_type pid, const std::vector<Real> & unodal_received)
  {
    const std::size_t msg_size = unodal_received.size();
    mooseAssert(msg_size == _nodes_to_receive[pid].size(),
                "Message size, " << msg_size
                                 << ", incompatible with nodes_to_receive, which has size "
                                 << _nodes_to_receive[pid].size());
    for (unsigned i = 0; i < msg_size; ++i)
      _u_nodal[_nodes_to_receive[pid][i]] = unodal_received[i];
  };
  Parallel::push_parallel_vector_data(this->comm(), unodal_to_send, unodal_action_functor);
 
  // Exchange _kij information with other processors
  std::map<processor_id_type, std::vector<Real>> kij_to_send;
  for (const auto & kv : _pairs_to_send)
  {
    const processor_id_type pid = kv.first;
    kij_to_send[pid] = std::vector<Real>();
    for (const auto & pr : kv.second)
      kij_to_send[pid].push_back(_kij[pr.first][pr.second]);
  }
 
  auto kij_action_functor = [this](processor_id_type pid, const std::vector<Real> & kij_received)
  {
    const std::size_t msg_size = kij_received.size();
    mooseAssert(msg_size == _pairs_to_receive[pid].size(),
                "Message size, " << msg_size
                                 << ", incompatible with pairs_to_receive, which has size "
                                 << _pairs_to_receive[pid].size());
    for (unsigned i = 0; i < msg_size; ++i)
      _kij[_pairs_to_receive[pid][i].first][_pairs_to_receive[pid][i].second] += kij_received[i];
  };
  Parallel::push_parallel_vector_data(this->comm(), kij_to_send, kij_action_functor);
}