#include <scheme/AcousticSolver.hpp>

#include <analysis/Polynomial1D.hpp>
#include <analysis/Polynomial1DRootComputer.hpp>
#include <language/utils/InterpolateItemValue.hpp>
#include <mesh/ItemValueUtils.hpp>
#include <mesh/MeshNodeBoundary.hpp>
#include <scheme/DirichletBoundaryConditionDescriptor.hpp>
#include <scheme/DiscreteFunctionP0.hpp>
#include <scheme/DiscreteFunctionUtils.hpp>
#include <scheme/IBoundaryConditionDescriptor.hpp>
#include <scheme/IDiscreteFunction.hpp>
#include <scheme/IDiscreteFunctionDescriptor.hpp>
#include <scheme/SymmetryBoundaryConditionDescriptor.hpp>

#include <variant>
#include <vector>

template <size_t Dimension>
double
acoustic_dt(const DiscreteFunctionP0<Dimension, double>& c)
{
  const std::shared_ptr<const Mesh<Connectivity<Dimension>>> mesh =
    std::dynamic_pointer_cast<const Mesh<Connectivity<Dimension>>>(c.mesh());

  const auto Vj = MeshDataManager::instance().getMeshData(*mesh).Vj();
  const auto Sj = MeshDataManager::instance().getMeshData(*mesh).sumOverRLjr();

  CellValue<double> local_dt{mesh->connectivity()};
  parallel_for(
    mesh->numberOfCells(), PUGS_LAMBDA(CellId j) { local_dt[j] = 2 * Vj[j] / (Sj[j] * c[j]); });

  return min(local_dt);
}

double
acoustic_dt(const std::shared_ptr<const IDiscreteFunction>& c)
{
  if ((c->descriptor().type() != DiscreteFunctionType::P0) or (c->dataType() != ASTNodeDataType::double_t)) {
    throw NormalError("invalid discrete function type");
  }

  std::shared_ptr mesh = c->mesh();

  switch (mesh->dimension()) {
  case 1: {
    return acoustic_dt(dynamic_cast<const DiscreteFunctionP0<1, double>&>(*c));
  }
  case 2: {
    return acoustic_dt(dynamic_cast<const DiscreteFunctionP0<2, double>&>(*c));
  }
  case 3: {
    return acoustic_dt(dynamic_cast<const DiscreteFunctionP0<3, double>&>(*c));
  }
  default: {
    throw UnexpectedError("invalid mesh dimension");
  }
  }
}

template <size_t Dimension>
class AcousticSolverHandler::AcousticSolver final : public AcousticSolverHandler::IAcousticSolver
{
 private:
  using Rdxd = TinyMatrix<Dimension>;
  using Rd   = TinyVector<Dimension>;

  using MeshType     = Mesh<Connectivity<Dimension>>;
  using MeshDataType = MeshData<Dimension>;

  using DiscreteScalarFunction = DiscreteFunctionP0<Dimension, double>;
  using DiscreteVectorFunction = DiscreteFunctionP0<Dimension, Rd>;

  class PressureBoundaryCondition;
  class SymmetryBoundaryCondition;
  class VelocityBoundaryCondition;

  using BoundaryCondition =
    std::variant<PressureBoundaryCondition, SymmetryBoundaryCondition, VelocityBoundaryCondition>;

  using BoundaryConditionList = std::vector<BoundaryCondition>;

  const SolverType m_solver_type;
  BoundaryConditionList m_boundary_condition_list;

  NodeValue<const Rd> m_ur;
  NodeValuePerCell<const Rd> m_Fjr;

  NodeValuePerCell<const Rdxd> m_Ajr;

  CellValue<const double>
  _getRhoC(const DiscreteScalarFunction& rho, const DiscreteScalarFunction& c)
  {
    Assert(rho.mesh() == c.mesh());

    std::shared_ptr mesh = std::dynamic_pointer_cast<const MeshType>(rho.mesh());

    CellValue<double> rhoc{mesh->connectivity()};
    parallel_for(
      mesh->numberOfCells(), PUGS_LAMBDA(CellId cell_id) { rhoc[cell_id] = rho[cell_id] * c[cell_id]; });

    return rhoc;
  }

  NodeValuePerCell<const Rdxd>
  _computeGlaceAjr(const MeshType& mesh, const CellValue<const double>& rhoc)
  {
    MeshDataType& mesh_data = MeshDataManager::instance().getMeshData(mesh);

    const NodeValuePerCell<const TinyVector<Dimension>> Cjr = mesh_data.Cjr();
    const NodeValuePerCell<const TinyVector<Dimension>> njr = mesh_data.njr();

    NodeValuePerCell<Rdxd> Ajr{mesh.connectivity()};

    parallel_for(
      mesh.numberOfCells(), PUGS_LAMBDA(CellId j) {
        const size_t& nb_nodes = Ajr.numberOfSubValues(j);
        const double& rhoc_j   = rhoc[j];
        for (size_t r = 0; r < nb_nodes; ++r) {
          Ajr(j, r) = tensorProduct(rhoc_j * Cjr(j, r), njr(j, r));
        }
      });

    return Ajr;
  }

  NodeValuePerCell<const Rdxd>
  _computeEucclhydAjr(const MeshType& mesh, const CellValue<const double>& rhoc)
  {
    MeshDataType& mesh_data = MeshDataManager::instance().getMeshData(mesh);

    const NodeValuePerFace<const Rd> Nlr = mesh_data.Nlr();
    const NodeValuePerFace<const Rd> nlr = mesh_data.nlr();

    const auto& face_to_node_matrix = mesh.connectivity().faceToNodeMatrix();
    const auto& cell_to_node_matrix = mesh.connectivity().cellToNodeMatrix();
    const auto& cell_to_face_matrix = mesh.connectivity().cellToFaceMatrix();

    NodeValuePerCell<Rdxd> Ajr{mesh.connectivity()};

    parallel_for(
      Ajr.numberOfValues(), PUGS_LAMBDA(size_t jr) { Ajr[jr] = zero; });

    parallel_for(
      mesh.numberOfCells(), PUGS_LAMBDA(CellId j) {
        const auto& cell_nodes = cell_to_node_matrix[j];

        const auto& cell_faces = cell_to_face_matrix[j];

        const double& rho_c = rhoc[j];

        for (size_t L = 0; L < cell_faces.size(); ++L) {
          const FaceId& l        = cell_faces[L];
          const auto& face_nodes = face_to_node_matrix[l];

          auto local_node_number_in_cell = [&](NodeId node_number) {
            for (size_t i_node = 0; i_node < cell_nodes.size(); ++i_node) {
              if (node_number == cell_nodes[i_node]) {
                return i_node;
              }
            }
            return std::numeric_limits<size_t>::max();
          };

          for (size_t rl = 0; rl < face_nodes.size(); ++rl) {
            const size_t R = local_node_number_in_cell(face_nodes[rl]);
            Ajr(j, R) += tensorProduct(rho_c * Nlr(l, rl), nlr(l, rl));
          }
        }
      });

    return Ajr;
  }

  NodeValuePerCell<const Rdxd>
  _computeAjr(const MeshType& mesh, const CellValue<const double>& rhoc)
  {
    if constexpr (Dimension == 1) {
      return _computeGlaceAjr(mesh, rhoc);
    } else {
      switch (m_solver_type) {
      case SolverType::Glace: {
        return _computeGlaceAjr(mesh, rhoc);
      }
      case SolverType::Eucclhyd: {
        return _computeEucclhydAjr(mesh, rhoc);
      }
      default: {
        throw UnexpectedError("invalid solver type");
      }
      }
    }
  }

  NodeValue<Rdxd>
  _computeAr(const MeshType& mesh, const NodeValuePerCell<const Rdxd>& Ajr)
  {
    const auto& node_to_cell_matrix               = mesh.connectivity().nodeToCellMatrix();
    const auto& node_local_numbers_in_their_cells = mesh.connectivity().nodeLocalNumbersInTheirCells();

    NodeValue<Rdxd> Ar{mesh.connectivity()};

    parallel_for(
      mesh.numberOfNodes(), PUGS_LAMBDA(NodeId r) {
        Rdxd sum                                   = zero;
        const auto& node_to_cell                   = node_to_cell_matrix[r];
        const auto& node_local_number_in_its_cells = node_local_numbers_in_their_cells.itemValues(r);

        for (size_t j = 0; j < node_to_cell.size(); ++j) {
          const CellId J       = node_to_cell[j];
          const unsigned int R = node_local_number_in_its_cells[j];
          sum += Ajr(J, R);
        }
        Ar[r] = sum;
      });

    return Ar;
  }

  NodeValue<Rd>
  _computeBr(const Mesh<Connectivity<Dimension>>& mesh,
             const NodeValuePerCell<const Rdxd>& Ajr,
             const DiscreteVectorFunction& u,
             const DiscreteScalarFunction& p)
  {
    MeshDataType& mesh_data = MeshDataManager::instance().getMeshData(mesh);

    const NodeValuePerCell<const Rd>& Cjr = mesh_data.Cjr();

    const auto& node_to_cell_matrix               = mesh.connectivity().nodeToCellMatrix();
    const auto& node_local_numbers_in_their_cells = mesh.connectivity().nodeLocalNumbersInTheirCells();

    NodeValue<Rd> b{mesh.connectivity()};

    parallel_for(
      mesh.numberOfNodes(), PUGS_LAMBDA(NodeId r) {
        const auto& node_to_cell                   = node_to_cell_matrix[r];
        const auto& node_local_number_in_its_cells = node_local_numbers_in_their_cells.itemValues(r);

        Rd br = zero;
        for (size_t j = 0; j < node_to_cell.size(); ++j) {
          const CellId J       = node_to_cell[j];
          const unsigned int R = node_local_number_in_its_cells[j];
          br += Ajr(J, R) * u[J] + p[J] * Cjr(J, R);
        }

        b[r] = br;
      });

    return b;
  }

  BoundaryConditionList
  _getBCList(const std::shared_ptr<const MeshType>& mesh,
             const std::vector<std::shared_ptr<const IBoundaryConditionDescriptor>>& bc_descriptor_list)
  {
    BoundaryConditionList bc_list;

    constexpr ItemType FaceType = [] {
      if constexpr (Dimension > 1) {
        return ItemType::face;
      } else {
        return ItemType::node;
      }
    }();

    for (const auto& bc_descriptor : bc_descriptor_list) {
      bool is_valid_boundary_condition = true;

      switch (bc_descriptor->type()) {
      case IBoundaryConditionDescriptor::Type::symmetry: {
        const SymmetryBoundaryConditionDescriptor& sym_bc_descriptor =
          dynamic_cast<const SymmetryBoundaryConditionDescriptor&>(*bc_descriptor);
        for (size_t i_ref_face_list = 0;
             i_ref_face_list < mesh->connectivity().template numberOfRefItemList<FaceType>(); ++i_ref_face_list) {
          const auto& ref_face_list = mesh->connectivity().template refItemList<FaceType>(i_ref_face_list);
          const RefId& ref          = ref_face_list.refId();
          if (ref == sym_bc_descriptor.boundaryDescriptor()) {
            SymmetryBoundaryCondition{MeshFlatNodeBoundary<Dimension>(mesh, ref_face_list)};
            bc_list.push_back(SymmetryBoundaryCondition{MeshFlatNodeBoundary<Dimension>(mesh, ref_face_list)});
          }
        }
        is_valid_boundary_condition = true;
        break;
      }

      case IBoundaryConditionDescriptor::Type::dirichlet: {
        const DirichletBoundaryConditionDescriptor& dirichlet_bc_descriptor =
          dynamic_cast<const DirichletBoundaryConditionDescriptor&>(*bc_descriptor);
        if (dirichlet_bc_descriptor.name() == "velocity") {
          for (size_t i_ref_face_list = 0;
               i_ref_face_list < mesh->connectivity().template numberOfRefItemList<FaceType>(); ++i_ref_face_list) {
            const auto& ref_face_list = mesh->connectivity().template refItemList<FaceType>(i_ref_face_list);
            const RefId& ref          = ref_face_list.refId();
            if (ref == dirichlet_bc_descriptor.boundaryDescriptor()) {
              MeshNodeBoundary<Dimension> mesh_node_boundary{mesh, ref_face_list};

              const FunctionSymbolId velocity_id = dirichlet_bc_descriptor.rhsSymbolId();

              const auto& node_list = mesh_node_boundary.nodeList();

              Array<const TinyVector<Dimension>> value_list = InterpolateItemValue<TinyVector<Dimension>(
                TinyVector<Dimension>)>::template interpolate<ItemType::node>(velocity_id, mesh->xr(), node_list);

              bc_list.push_back(VelocityBoundaryCondition{node_list, value_list});
            }
          }
        } else if (dirichlet_bc_descriptor.name() == "pressure") {
          for (size_t i_ref_face_list = 0;
               i_ref_face_list < mesh->connectivity().template numberOfRefItemList<FaceType>(); ++i_ref_face_list) {
            const auto& ref_face_list = mesh->connectivity().template refItemList<FaceType>(i_ref_face_list);
            const RefId& ref          = ref_face_list.refId();
            if (ref == dirichlet_bc_descriptor.boundaryDescriptor()) {
              const auto& face_list = ref_face_list.list();

              const FunctionSymbolId pressure_id = dirichlet_bc_descriptor.rhsSymbolId();

              Array<const double> face_values = [&] {
                if constexpr (Dimension == 1) {
                  return InterpolateItemValue<double(
                    TinyVector<Dimension>)>::template interpolate<FaceType>(pressure_id, mesh->xr(), face_list);
                } else {
                  MeshDataType& mesh_data = MeshDataManager::instance().getMeshData(*mesh);

                  return InterpolateItemValue<double(
                    TinyVector<Dimension>)>::template interpolate<FaceType>(pressure_id, mesh_data.xl(), face_list);
                }
              }();

              bc_list.push_back(PressureBoundaryCondition{face_list, face_values});
            }
          }
        } else {
          is_valid_boundary_condition = false;
        }
        break;
      }
      default: {
        is_valid_boundary_condition = false;
      }
      }
      if (not is_valid_boundary_condition) {
        std::ostringstream error_msg;
        error_msg << *bc_descriptor << " is an invalid boundary condition for acoustic solver";
        throw NormalError(error_msg.str());
      }
    }

    return bc_list;
  }

  void _applyPressureBC(const MeshType& mesh, NodeValue<Rd>& br);
  void _applySymmetryBC(NodeValue<Rdxd>& Ar, NodeValue<Rd>& br);
  void _applyVelocityBC(NodeValue<Rdxd>& Ar, NodeValue<Rd>& br);

  void
  _applyBoundaryConditions(const MeshType& mesh, NodeValue<Rdxd>& Ar, NodeValue<Rd>& br)
  {
    this->_applyPressureBC(mesh, br);
    this->_applySymmetryBC(Ar, br);
    this->_applyVelocityBC(Ar, br);
  }

  NodeValue<const Rd>
  _computeUr(const MeshType& mesh, const NodeValue<Rdxd>& Ar, const NodeValue<Rd>& br)
  {
    NodeValue<Rd> u{mesh.connectivity()};
    parallel_for(
      mesh.numberOfNodes(), PUGS_LAMBDA(NodeId r) { u[r] = inverse(Ar[r]) * br[r]; });

    return u;
  }

  NodeValuePerCell<Rd>
  _computeFjr(const MeshType& mesh,
              const NodeValuePerCell<const Rdxd>& Ajr,
              const NodeValue<const Rd>& ur,
              const DiscreteVectorFunction& u,
              const DiscreteScalarFunction& p)
  {
    MeshDataType& mesh_data = MeshDataManager::instance().getMeshData(mesh);

    const NodeValuePerCell<const Rd> Cjr = mesh_data.Cjr();

    const auto& cell_to_node_matrix = mesh.connectivity().cellToNodeMatrix();

    NodeValuePerCell<Rd> F{mesh.connectivity()};
    parallel_for(
      mesh.numberOfCells(), PUGS_LAMBDA(CellId j) {
        const auto& cell_nodes = cell_to_node_matrix[j];

        for (size_t r = 0; r < cell_nodes.size(); ++r) {
          F(j, r) = Ajr(j, r) * (u[j] - ur[cell_nodes[r]]) + p[j] * Cjr(j, r);
        }
      });

    return F;
  }

  double
  _computeMaxDensityDt(const MeshType& mesh,
                       const DiscreteFunctionP0<Dimension, double>& rho,
                       const double dt_max) const
  {
    double dt = dt_max;

    MeshDataType& mesh_data = MeshDataManager::instance().getMeshData(mesh);

    const NodeValuePerCell<const Rd> Cjr = mesh_data.Cjr();
    const CellValue<const double> Vj     = mesh_data.Vj();

    const auto& cell_to_node_matrix = mesh.connectivity().cellToNodeMatrix();

    CellValue<double> dVj{mesh.connectivity()};

    parallel_for(
      mesh.numberOfCells(), PUGS_LAMBDA(CellId j) {
        const auto& cell_nodes = cell_to_node_matrix[j];

        double dV = 0;

        for (size_t R = 0; R < cell_nodes.size(); ++R) {
          const NodeId r = cell_nodes[R];
          dV += dot(Cjr(j, R), m_ur[r]);
        }

        dVj[j] = dV;
      });

    CellValue<double> Gj{mesh.connectivity()};

    if constexpr (Dimension == 2) {
      for (CellId j = 0; j < mesh.numberOfCells(); ++j) {
        const auto& cell_nodes = cell_to_node_matrix[j];

        double G = 0;
        for (size_t R = 0; R < cell_nodes.size(); ++R) {
          const NodeId r   = cell_nodes[R];
          const NodeId rp1 = cell_nodes[(R + 1) % cell_nodes.size()];
          G -= dot(m_ur[rp1], Rd{m_ur[r][1], -m_ur[r][0]});
        }
        Gj[j] = 0.5 * G;
      }
    }

    for (CellId j = 0; j < mesh.numberOfCells(); ++j) {
      Polynomial1D p_tau({1. / rho[j], dVj[j] / (rho[j] * Vj[j])});
      if constexpr (Dimension == 2) {
        p_tau = p_tau + Polynomial1D({0, 0, Gj[j] / (rho[j] * Vj[j])});
      }
      Polynomial1DRootComputer computer{p_tau, 0, dt};
      std::optional dt_tau = computer.getFirstRoot();
      if (dt_tau.has_value()) {
        Assert(dt_tau.value() <= dt);
        dt = dt_tau.value();
      }
    }

    if (dt != dt_max) {
      std::cout << "volume variation imposes time step\n";
    }

    if (dt < dt_max) {
      dt *= 0.95 * dt;
    }

    return parallel::allReduceMin(dt);
  }

  double
  _computeMaxEnergyDt(const MeshType& mesh,
                      const DiscreteScalarFunction& rho,
                      const DiscreteVectorFunction& u,
                      const DiscreteScalarFunction& E,
                      const DiscreteScalarFunction& p,
                      const double dt_max) const
  {
    double dt = dt_max;

    MeshDataType& mesh_data = MeshDataManager::instance().getMeshData(mesh);

    const NodeValuePerCell<const Rd> Cjr = mesh_data.Cjr();
    const CellValue<const double> Vj     = mesh_data.Vj();

    const auto& cell_to_node_matrix = mesh.connectivity().cellToNodeMatrix();

    CellValue<double> dVj{mesh.connectivity()};

    parallel_for(
      mesh.numberOfCells(), PUGS_LAMBDA(CellId j) {
        const auto& cell_nodes = cell_to_node_matrix[j];

        double dV = 0;

        for (size_t R = 0; R < cell_nodes.size(); ++R) {
          const NodeId r = cell_nodes[R];
          dV += dot(Cjr(j, R), m_ur[r]);
        }

        dVj[j] = dV;
      });

    CellValue<double> dSj{mesh.connectivity()};

    parallel_for(
      mesh.numberOfCells(), PUGS_LAMBDA(CellId j) {
        const auto& cell_nodes = cell_to_node_matrix[j];

        double dS = 0;

        for (size_t R = 0; R < cell_nodes.size(); ++R) {
          const NodeId r = cell_nodes[R];

          Rd du = u[j] - m_ur[r];
          dS += dot(m_Ajr(j, R) * du, du);
        }

        dSj[j] = dS;
      });

    CellValue<Rd> dPj{mesh.connectivity()};

    parallel_for(
      mesh.numberOfCells(), PUGS_LAMBDA(CellId j) {
        const auto& cell_nodes = cell_to_node_matrix[j];

        Rd dP = zero;

        for (size_t R = 0; R < cell_nodes.size(); ++R) {
          const NodeId r = cell_nodes[R];

          Rd du = u[j] - m_ur[r];
          dP += m_Ajr(j, R) * du;
        }

        dPj[j] = dP;
      });

    for (CellId j = 0; j < mesh.numberOfCells(); ++j) {
      const double inv_mj = 1 / (rho[j] * Vj[j]);

      Polynomial1D p_e(
        {E[j] - 0.5 * dot(u[j], u[j]), inv_mj * (dSj[j] - p[j] * dVj[j]), -inv_mj * inv_mj * dot(dPj[j], dPj[j])});
      Polynomial1DRootComputer computer{p_e, 0, dt};
      std::optional dt_e = computer.getFirstRoot();
      if (dt_e.has_value()) {
        Assert(dt_e.value() <= dt);
        dt = dt_e.value();
      }
    }

    if (dt < dt_max) {
      dt *= 0.95 * dt;
    }

    if (dt != dt_max) {
      std::cout << "internal energy variation imposes time step\n";
    }

    return parallel::allReduceMin(dt);
  }

  double
  _computeMaxEntropyDt(const MeshType& mesh,
                       const DiscreteScalarFunction& rho,
                       const DiscreteVectorFunction& u,
                       const DiscreteScalarFunction& p,
                       const double dt_max) const
  {
    double dt = dt_max;

    MeshDataType& mesh_data = MeshDataManager::instance().getMeshData(mesh);

    const NodeValuePerCell<const Rd> Cjr = mesh_data.Cjr();
    const CellValue<const double> Vj     = mesh_data.Vj();

    const auto& cell_to_node_matrix = mesh.connectivity().cellToNodeMatrix();

    CellValue<double> dVj{mesh.connectivity()};

    parallel_for(
      mesh.numberOfCells(), PUGS_LAMBDA(CellId j) {
        const auto& cell_nodes = cell_to_node_matrix[j];

        double dV = 0;

        for (size_t R = 0; R < cell_nodes.size(); ++R) {
          const NodeId r = cell_nodes[R];
          dV += dot(Cjr(j, R), m_ur[r]);
        }

        dVj[j] = dV;
      });

    CellValue<double> Gj{mesh.connectivity()};

    if constexpr (Dimension == 2) {
      for (CellId j = 0; j < mesh.numberOfCells(); ++j) {
        const auto& cell_nodes = cell_to_node_matrix[j];

        double G = 0;
        for (size_t R = 0; R < cell_nodes.size(); ++R) {
          const NodeId r   = cell_nodes[R];
          const NodeId rp1 = cell_nodes[(R + 1) % cell_nodes.size()];
          G -= dot(m_ur[rp1], Rd{m_ur[r][1], -m_ur[r][0]});
        }
        Gj[j] = 0.5 * G;
      }
    }

    CellValue<double> dSj{mesh.connectivity()};

    parallel_for(
      mesh.numberOfCells(), PUGS_LAMBDA(CellId j) {
        const auto& cell_nodes = cell_to_node_matrix[j];

        double dS = 0;

        for (size_t R = 0; R < cell_nodes.size(); ++R) {
          const NodeId r = cell_nodes[R];

          const Rd du = u[j] - m_ur[r];
          dS += dot(m_Ajr(j, R) * du, du);
        }

        dSj[j] = dS;
      });

    CellValue<Rd> dPj{mesh.connectivity()};

    parallel_for(
      mesh.numberOfCells(), PUGS_LAMBDA(CellId j) {
        const auto& cell_nodes = cell_to_node_matrix[j];

        Rd dP = zero;

        for (size_t R = 0; R < cell_nodes.size(); ++R) {
          const NodeId r = cell_nodes[R];

          Rd du = u[j] - m_ur[r];
          dP += m_Ajr(j, R) * du;
        }

        dPj[j] = dP;
      });

#warning fixed gamma value
    const double gamma = 1.4;

    for (CellId j = 0; j < mesh.numberOfCells(); ++j) {
      const double inv_mj = 1 / (rho[j] * Vj[j]);
      const double inv_Vj = 1 / Vj[j];

      Polynomial1D DT({0, 1});
      std::vector<double> delta_S = {dSj[j], -0.5 * inv_mj * dot(dPj[j], dPj[j])};
      if constexpr (Dimension == 2) {
        delta_S[1] += p[j] * Gj[j];
      }
      Polynomial1D p_S0(delta_S);

      std::vector<double> delta_V{dVj[j] /*, dG */};
      if constexpr (Dimension == 2) {
        delta_V.push_back(Gj[j]);
      }
      Polynomial1D p_deltaV(delta_V);

      Polynomial1D p_S1({dSj[j] - p[j] * dVj[j], -0.5 * inv_mj * dot(dPj[j], dPj[j])});

      Polynomial1D p_S =                                                                                 //
        p_S0                                                                                             //
        + (inv_Vj * DT) * (((gamma - 1) * p_S1) * p_deltaV + (gamma - 2) * p[j] * p_deltaV * p_deltaV)   //
        + (inv_Vj * DT) * (inv_Vj * DT) * (((gamma - 1) * (gamma - 2)) * p_S1) * p_deltaV * p_deltaV;

      Polynomial1DRootComputer computer{p_S, 0, dt};
      std::optional dt_S = computer.getFirstRoot();
      if (dt_S.has_value()) {
        Assert(dt_S.value() <= dt);
        dt = dt_S.value();
      }
    }

    if (dt != dt_max) {
      std::cout << "entropy variation imposes time step\n";
    }
    return parallel::allReduceMin(dt);
  }

  AcousticSolver(SolverType solver_type,
                 const std::shared_ptr<const MeshType>& p_mesh,
                 const DiscreteScalarFunction& rho,
                 const DiscreteScalarFunction& c,
                 const DiscreteVectorFunction& u,
                 const DiscreteScalarFunction& p,
                 const std::vector<std::shared_ptr<const IBoundaryConditionDescriptor>>& bc_descriptor_list)
    : m_solver_type{solver_type}, m_boundary_condition_list{this->_getBCList(p_mesh, bc_descriptor_list)}
  {
    const MeshType& mesh = *p_mesh;

    CellValue<const double> rhoc     = this->_getRhoC(rho, c);
    NodeValuePerCell<const Rdxd> Ajr = this->_computeAjr(mesh, rhoc);

    m_Ajr = Ajr;

    NodeValue<Rdxd> Ar = this->_computeAr(mesh, Ajr);
    NodeValue<Rd> br   = this->_computeBr(mesh, Ajr, u, p);

    this->_applyBoundaryConditions(mesh, Ar, br);

    synchronize(Ar);
    synchronize(br);

    m_ur  = this->_computeUr(mesh, Ar, br);
    m_Fjr = this->_computeFjr(mesh, Ajr, m_ur, u, p);
  }

 public:
  std::tuple<std::shared_ptr<const IMesh>,
             std::shared_ptr<const DiscreteFunctionP0<Dimension, double>>,
             std::shared_ptr<const DiscreteFunctionP0<Dimension, Rd>>,
             std::shared_ptr<const DiscreteFunctionP0<Dimension, double>>>
  apply(const double& dt,
        const std::shared_ptr<const MeshType>& mesh,
        const DiscreteFunctionP0<Dimension, double>& rho,
        const DiscreteFunctionP0<Dimension, Rd>& u,
        const DiscreteFunctionP0<Dimension, double>& E) const
  {
    const auto& cell_to_node_matrix = mesh->connectivity().cellToNodeMatrix();

    NodeValue<Rd> new_xr = copy(mesh->xr());
    parallel_for(
      mesh->numberOfNodes(), PUGS_LAMBDA(NodeId r) { new_xr[r] += dt * m_ur[r]; });

    std::shared_ptr<const MeshType> new_mesh = std::make_shared<MeshType>(mesh->shared_connectivity(), new_xr);

    CellValue<const double> Vj = MeshDataManager::instance().getMeshData(*mesh).Vj();

    CellValue<double> new_rho = copy(rho.cellValues());
    CellValue<Rd> new_u       = copy(u.cellValues());
    CellValue<double> new_E   = copy(E.cellValues());

    parallel_for(
      mesh->numberOfCells(), PUGS_LAMBDA(CellId j) {
        const auto& cell_nodes = cell_to_node_matrix[j];

        Rd momentum_fluxes   = zero;
        double energy_fluxes = 0;
        for (size_t R = 0; R < cell_nodes.size(); ++R) {
          const NodeId r = cell_nodes[R];
          momentum_fluxes += m_Fjr(j, R);
          energy_fluxes += dot(m_Fjr(j, R), m_ur[r]);
        }
        const double dt_over_Mj = dt / (rho[j] * Vj[j]);
        new_u[j] -= dt_over_Mj * momentum_fluxes;
        new_E[j] -= dt_over_Mj * energy_fluxes;
      });

    CellValue<const double> new_Vj = MeshDataManager::instance().getMeshData(*new_mesh).Vj();

    parallel_for(
      mesh->numberOfCells(), PUGS_LAMBDA(CellId j) { new_rho[j] *= Vj[j] / new_Vj[j]; });

    return {new_mesh, std::make_shared<DiscreteScalarFunction>(new_mesh, new_rho),
            std::make_shared<DiscreteVectorFunction>(new_mesh, new_u),
            std::make_shared<DiscreteScalarFunction>(new_mesh, new_E)};
  }

  std::tuple<std::shared_ptr<const IMesh>,
             std::shared_ptr<const IDiscreteFunction>,
             std::shared_ptr<const IDiscreteFunction>,
             std::shared_ptr<const IDiscreteFunction>>
  apply(const double& dt,
        const std::shared_ptr<const IDiscreteFunction>& rho,
        const std::shared_ptr<const IDiscreteFunction>& u,
        const std::shared_ptr<const IDiscreteFunction>& E) const
  {
    std::shared_ptr i_mesh = getCommonMesh({rho, u, E});
    if (not i_mesh) {
      throw NormalError("discrete functions are not defined on the same mesh");
    }

    if (not checkDiscretizationType({rho, u, E}, DiscreteFunctionType::P0)) {
      throw NormalError("acoustic solver expects P0 functions");
    }

    return this->apply(dt, std::dynamic_pointer_cast<const MeshType>(i_mesh),
                       *std::dynamic_pointer_cast<const DiscreteScalarFunction>(rho),
                       *std::dynamic_pointer_cast<const DiscreteVectorFunction>(u),
                       *std::dynamic_pointer_cast<const DiscreteScalarFunction>(E));
  }

  std::tuple<double,
             std::shared_ptr<const IMesh>,
             std::shared_ptr<const DiscreteScalarFunction>,
             std::shared_ptr<const DiscreteVectorFunction>,
             std::shared_ptr<const DiscreteScalarFunction>>
  apply_computing_entropic_dt(const double& dt_max,
                              const std::shared_ptr<const MeshType>& mesh,
                              const DiscreteScalarFunction& rho,
                              const DiscreteVectorFunction& u,
                              const DiscreteScalarFunction& E,
                              const DiscreteScalarFunction& p) const
  {
    const auto& cell_to_node_matrix = mesh->connectivity().cellToNodeMatrix();

    double dt = dt_max;

    dt = _computeMaxDensityDt(*mesh, rho, dt);
    dt = _computeMaxEnergyDt(*mesh, rho, u, E, p, dt);
    dt = _computeMaxEntropyDt(*mesh, rho, u, p, dt);

    NodeValue<Rd> new_xr = copy(mesh->xr());
    parallel_for(
      mesh->numberOfNodes(), PUGS_LAMBDA(NodeId r) { new_xr[r] += dt * m_ur[r]; });

    std::shared_ptr<const MeshType> new_mesh = std::make_shared<MeshType>(mesh->shared_connectivity(), new_xr);

    CellValue<const double> Vj = MeshDataManager::instance().getMeshData(*mesh).Vj();

    CellValue<double> new_rho = copy(rho.cellValues());
    CellValue<Rd> new_u       = copy(u.cellValues());
    CellValue<double> new_E   = copy(E.cellValues());

    parallel_for(
      mesh->numberOfCells(), PUGS_LAMBDA(CellId j) {
        const auto& cell_nodes = cell_to_node_matrix[j];

        Rd momentum_fluxes   = zero;
        double energy_fluxes = 0;
        for (size_t R = 0; R < cell_nodes.size(); ++R) {
          const NodeId r = cell_nodes[R];
          momentum_fluxes += m_Fjr(j, R);
          energy_fluxes += dot(m_Fjr(j, R), m_ur[r]);
        }
        const double dt_over_Mj = dt / (rho[j] * Vj[j]);
        new_u[j] -= dt_over_Mj * momentum_fluxes;
        new_E[j] -= dt_over_Mj * energy_fluxes;
      });

    CellValue<const double> new_Vj = MeshDataManager::instance().getMeshData(*new_mesh).Vj();

    parallel_for(
      mesh->numberOfCells(), PUGS_LAMBDA(CellId j) { new_rho[j] *= Vj[j] / new_Vj[j]; });

    return {dt, new_mesh, std::make_shared<DiscreteScalarFunction>(new_mesh, new_rho),
            std::make_shared<DiscreteVectorFunction>(new_mesh, new_u),
            std::make_shared<DiscreteScalarFunction>(new_mesh, new_E)};
  }

  std::tuple<double,
             std::shared_ptr<const IMesh>,
             std::shared_ptr<const IDiscreteFunction>,
             std::shared_ptr<const IDiscreteFunction>,
             std::shared_ptr<const IDiscreteFunction>>
  apply_computing_entropic_dt(const double& dt_max,
                              const std::shared_ptr<const IDiscreteFunction>& rho,
                              const std::shared_ptr<const IDiscreteFunction>& u,
                              const std::shared_ptr<const IDiscreteFunction>& E,
                              const std::shared_ptr<const IDiscreteFunction>& p) const
  {
    std::shared_ptr i_mesh = getCommonMesh({rho, u, E});
    if (not i_mesh) {
      throw NormalError("discrete functions are not defined on the same mesh");
    }

    if (not checkDiscretizationType({rho, u, E}, DiscreteFunctionType::P0)) {
      throw NormalError("acoustic solver expects P0 functions");
    }

    return this->apply_computing_entropic_dt(dt_max, std::dynamic_pointer_cast<const MeshType>(i_mesh),
                                             *std::dynamic_pointer_cast<const DiscreteScalarFunction>(rho),
                                             *std::dynamic_pointer_cast<const DiscreteVectorFunction>(u),
                                             *std::dynamic_pointer_cast<const DiscreteScalarFunction>(E),
                                             *std::dynamic_pointer_cast<const DiscreteScalarFunction>(p));
  }

  AcousticSolver(SolverType solver_type,
                 const std::shared_ptr<const IMesh>& mesh,
                 const std::shared_ptr<const IDiscreteFunction>& rho,
                 const std::shared_ptr<const IDiscreteFunction>& c,
                 const std::shared_ptr<const IDiscreteFunction>& u,
                 const std::shared_ptr<const IDiscreteFunction>& p,
                 const std::vector<std::shared_ptr<const IBoundaryConditionDescriptor>>& bc_descriptor_list)
    : AcousticSolver{solver_type,
                     std::dynamic_pointer_cast<const Mesh<Connectivity<Dimension>>>(mesh),
                     *std::dynamic_pointer_cast<const DiscreteScalarFunction>(rho),
                     *std::dynamic_pointer_cast<const DiscreteScalarFunction>(c),
                     *std::dynamic_pointer_cast<const DiscreteVectorFunction>(u),
                     *std::dynamic_pointer_cast<const DiscreteScalarFunction>(p),
                     bc_descriptor_list}
  {}

  AcousticSolver()                 = default;
  AcousticSolver(AcousticSolver&&) = default;
  ~AcousticSolver()                = default;
};

template <size_t Dimension>
void
AcousticSolverHandler::AcousticSolver<Dimension>::_applyPressureBC(const MeshType& mesh, NodeValue<Rd>& br)
{
  for (const auto& boundary_condition : m_boundary_condition_list) {
    std::visit(
      [&](auto&& bc) {
        using T = std::decay_t<decltype(bc)>;
        if constexpr (std::is_same_v<PressureBoundaryCondition, T>) {
          MeshData<Dimension>& mesh_data = MeshDataManager::instance().getMeshData(mesh);
          if constexpr (Dimension == 1) {
            const NodeValuePerCell<const Rd> Cjr = mesh_data.Cjr();

            const auto& node_to_cell_matrix               = mesh.connectivity().nodeToCellMatrix();
            const auto& node_local_numbers_in_their_cells = mesh.connectivity().nodeLocalNumbersInTheirCells();

            const auto& node_list  = bc.faceList();
            const auto& value_list = bc.valueList();
            parallel_for(
              node_list.size(), PUGS_LAMBDA(size_t i_node) {
                const NodeId node_id       = node_list[i_node];
                const auto& node_cell_list = node_to_cell_matrix[node_id];
                Assert(node_cell_list.size() == 1);

                CellId node_cell_id              = node_cell_list[0];
                size_t node_local_number_in_cell = node_local_numbers_in_their_cells(node_id, 0);

                br[node_id] -= value_list[i_node] * Cjr(node_cell_id, node_local_number_in_cell);
              });
          } else {
            const NodeValuePerFace<const Rd> Nlr = mesh_data.Nlr();

            const auto& face_to_cell_matrix               = mesh.connectivity().faceToCellMatrix();
            const auto& face_to_node_matrix               = mesh.connectivity().faceToNodeMatrix();
            const auto& face_local_numbers_in_their_cells = mesh.connectivity().faceLocalNumbersInTheirCells();
            const auto& face_cell_is_reversed             = mesh.connectivity().cellFaceIsReversed();

            const auto& face_list  = bc.faceList();
            const auto& value_list = bc.valueList();
            for (size_t i_face = 0; i_face < face_list.size(); ++i_face) {
              const FaceId face_id       = face_list[i_face];
              const auto& face_cell_list = face_to_cell_matrix[face_id];
              Assert(face_cell_list.size() == 1);

              CellId face_cell_id              = face_cell_list[0];
              size_t face_local_number_in_cell = face_local_numbers_in_their_cells(face_id, 0);

              const double sign = face_cell_is_reversed(face_cell_id, face_local_number_in_cell) ? -1 : 1;

              const auto& face_nodes = face_to_node_matrix[face_id];

              for (size_t i_node = 0; i_node < face_nodes.size(); ++i_node) {
                NodeId node_id = face_nodes[i_node];
                br[node_id] -= sign * value_list[i_face] * Nlr(face_id, i_node);
              }
            }
          }
        }
      },
      boundary_condition);
  }
}

template <size_t Dimension>
void
AcousticSolverHandler::AcousticSolver<Dimension>::_applySymmetryBC(NodeValue<Rdxd>& Ar, NodeValue<Rd>& br)
{
  for (const auto& boundary_condition : m_boundary_condition_list) {
    std::visit(
      [&](auto&& bc) {
        using T = std::decay_t<decltype(bc)>;
        if constexpr (std::is_same_v<SymmetryBoundaryCondition, T>) {
          const Rd& n = bc.outgoingNormal();

          const Rdxd I   = identity;
          const Rdxd nxn = tensorProduct(n, n);
          const Rdxd P   = I - nxn;

          const Array<const NodeId>& node_list = bc.nodeList();
          parallel_for(
            bc.numberOfNodes(), PUGS_LAMBDA(int r_number) {
              const NodeId r = node_list[r_number];

              Ar[r] = P * Ar[r] * P + nxn;
              br[r] = P * br[r];
            });
        }
      },
      boundary_condition);
  }
}

template <size_t Dimension>
void
AcousticSolverHandler::AcousticSolver<Dimension>::_applyVelocityBC(NodeValue<Rdxd>& Ar, NodeValue<Rd>& br)
{
  for (const auto& boundary_condition : m_boundary_condition_list) {
    std::visit(
      [&](auto&& bc) {
        using T = std::decay_t<decltype(bc)>;
        if constexpr (std::is_same_v<VelocityBoundaryCondition, T>) {
          const auto& node_list  = bc.nodeList();
          const auto& value_list = bc.valueList();

          parallel_for(
            node_list.size(), PUGS_LAMBDA(size_t i_node) {
              NodeId node_id    = node_list[i_node];
              const auto& value = value_list[i_node];

              Ar[node_id] = identity;
              br[node_id] = value;
            });
        }
      },
      boundary_condition);
  }
}

template <size_t Dimension>
class AcousticSolverHandler::AcousticSolver<Dimension>::PressureBoundaryCondition
{
 private:
  const Array<const double> m_value_list;
  const Array<const FaceId> m_face_list;

 public:
  const Array<const FaceId>&
  faceList() const
  {
    return m_face_list;
  }

  const Array<const double>&
  valueList() const
  {
    return m_value_list;
  }

  PressureBoundaryCondition(const Array<const FaceId>& face_list, const Array<const double>& value_list)
    : m_value_list{value_list}, m_face_list{face_list}
  {}

  ~PressureBoundaryCondition() = default;
};

template <>
class AcousticSolverHandler::AcousticSolver<1>::PressureBoundaryCondition
{
 private:
  const Array<const double> m_value_list;
  const Array<const NodeId> m_face_list;

 public:
  const Array<const NodeId>&
  faceList() const
  {
    return m_face_list;
  }

  const Array<const double>&
  valueList() const
  {
    return m_value_list;
  }

  PressureBoundaryCondition(const Array<const NodeId>& face_list, const Array<const double>& value_list)
    : m_value_list{value_list}, m_face_list{face_list}
  {}

  ~PressureBoundaryCondition() = default;
};

template <size_t Dimension>
class AcousticSolverHandler::AcousticSolver<Dimension>::VelocityBoundaryCondition
{
 private:
  const Array<const TinyVector<Dimension>> m_value_list;
  const Array<const NodeId> m_node_list;

 public:
  const Array<const NodeId>&
  nodeList() const
  {
    return m_node_list;
  }

  const Array<const TinyVector<Dimension>>&
  valueList() const
  {
    return m_value_list;
  }

  VelocityBoundaryCondition(const Array<const NodeId>& node_list, const Array<const TinyVector<Dimension>>& value_list)
    : m_value_list{value_list}, m_node_list{node_list}
  {}

  ~VelocityBoundaryCondition() = default;
};

template <size_t Dimension>
class AcousticSolverHandler::AcousticSolver<Dimension>::SymmetryBoundaryCondition
{
 public:
  using Rd = TinyVector<Dimension, double>;

 private:
  const MeshFlatNodeBoundary<Dimension> m_mesh_flat_node_boundary;

 public:
  const Rd&
  outgoingNormal() const
  {
    return m_mesh_flat_node_boundary.outgoingNormal();
  }

  size_t
  numberOfNodes() const
  {
    return m_mesh_flat_node_boundary.nodeList().size();
  }

  const Array<const NodeId>&
  nodeList() const
  {
    return m_mesh_flat_node_boundary.nodeList();
  }

  SymmetryBoundaryCondition(const MeshFlatNodeBoundary<Dimension>& mesh_flat_node_boundary)
    : m_mesh_flat_node_boundary(mesh_flat_node_boundary)
  {
    ;
  }

  ~SymmetryBoundaryCondition() = default;
};

AcousticSolverHandler::AcousticSolverHandler(
  SolverType solver_type,
  const std::shared_ptr<const IDiscreteFunction>& rho,
  const std::shared_ptr<const IDiscreteFunction>& c,
  const std::shared_ptr<const IDiscreteFunction>& u,
  const std::shared_ptr<const IDiscreteFunction>& p,
  const std::vector<std::shared_ptr<const IBoundaryConditionDescriptor>>& bc_descriptor_list)
{
  std::shared_ptr i_mesh = getCommonMesh({rho, c, u, p});
  if (not i_mesh) {
    throw NormalError("discrete functions are not defined on the same mesh");
  }

  if (not checkDiscretizationType({rho, c, u, p}, DiscreteFunctionType::P0)) {
    throw NormalError("acoustic solver expects P0 functions");
  }

  switch (i_mesh->dimension()) {
  case 1: {
    m_acoustic_solver = std::make_unique<AcousticSolver<1>>(solver_type, i_mesh, rho, c, u, p, bc_descriptor_list);
    break;
  }
  case 2: {
    m_acoustic_solver = std::make_unique<AcousticSolver<2>>(solver_type, i_mesh, rho, c, u, p, bc_descriptor_list);
    break;
  }
  case 3: {
    m_acoustic_solver = std::make_unique<AcousticSolver<3>>(solver_type, i_mesh, rho, c, u, p, bc_descriptor_list);
    break;
  }
  default: {
    throw UnexpectedError("invalid mesh dimension");
  }
  }
}