initial_state_becker_solution.h

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//
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
// Copyright (C) 2022 - 2024 by the ryujin authors
//
 
#pragma once
 
#include <initial_state_library.h>
 
namespace ryujin
{
  namespace EulerInitialStates
  {
    template <typename Description, int dim, typename Number>
    class BeckerSolution : public InitialState<Description, dim, Number>
    {
    public:
      using HyperbolicSystem = typename Description::HyperbolicSystem;
      using View =
          typename Description::template HyperbolicSystemView<dim, Number>;
      using state_type = typename View::state_type;
 
      BeckerSolution(const HyperbolicSystem &hyperbolic_system,
                     const std::string &subsection)
          : InitialState<Description, dim, Number>("becker solution",
                                                   subsection)
          , hyperbolic_system_(hyperbolic_system)
      {
        gamma_ = 1.4;
        if constexpr (!View::have_gamma) {
          this->add_parameter("gamma", gamma_, "The ratio of specific heats");
        }
 
        velocity_ = 0.2;
        this->add_parameter("velocity galilean frame",
                            velocity_,
                            "Velocity used to apply a Galilean transformation "
                            "to the otherwise stationary solution");
 
        velocity_left_ = 1.0;
        this->add_parameter(
            "velocity left", velocity_left_, "Left limit velocity");
 
        velocity_right_ = 7. / 27.;
        this->add_parameter(
            "velocity right", velocity_right_, "Right limit velocity");
 
        density_left_ = 1.0;
        this->add_parameter(
            "density left", density_left_, "Left limit density");
 
        mu_ = 0.01;
        this->add_parameter("mu", mu_, "Shear viscosity");
 
        /* Callback: */
 
        dealii::ParameterAcceptor::parse_parameters_call_back.connect([this]() {
          const auto view = hyperbolic_system_.template view<dim, Number>();
 
          if constexpr (View::have_gamma) {
            gamma_ = view.gamma();
          }
 
          AssertThrow(
              velocity_left_ > velocity_right_,
              dealii::ExcMessage("The left limiting velocity must be greater "
                                 "than the right limiting velocity"));
 
          AssertThrow(velocity_left_ > 0.,
                      dealii::ExcMessage(
                          "The left limiting velocity must be positive"));
 
          /*
           * Set up all helper functions and quantities:
           */
 
          const double velocity_origin =
              std::sqrt(velocity_left_ * velocity_right_);
 
          /* Prefactor as given in: (7.1) */
 
          const double Pr = 0.75;
          const double factor = 2. * gamma_ / (gamma_ + 1.) //
                                * mu_ / (density_left_ * velocity_left_ * Pr);
 
          psi = [=, this](double x, double v) {
            const double c_l =
                velocity_left_ / (velocity_left_ - velocity_right_);
            const double c_r =
                velocity_right_ / (velocity_left_ - velocity_right_);
            const double log_l = std::log(velocity_left_ - v) -
                                 std::log(velocity_left_ - velocity_origin);
            const double log_r = std::log(v - velocity_right_) -
                                 std::log(velocity_origin - velocity_right_);
 
            const double value = factor * (c_l * log_l - c_r * log_r) - x;
 
            const double derivative = factor * (-c_l / (velocity_left_ - v) -
                                                c_r / (v - velocity_right_));
 
            return std::make_tuple(value, derivative);
          };
 
          /* Determine cut-off points: */
 
          constexpr double tol = 1.e-12;
 
          const double x_left = std::get<0>(
              psi(0., (1. - tol) * velocity_left_ + tol * velocity_right_));
 
          const double x_right = std::get<0>(
              psi(0., tol * velocity_left_ + (1. - tol) * velocity_right_));
 
          const double norm = (x_right - x_left) * tol;
 
          /* Root finding algorithm: */
 
          find_velocity = [=, this](double x) {
            /* Return extremal cases: */
            if (x <= x_left)
              return double(velocity_left_);
            if (x >= x_right)
              return double(velocity_right_);
 
            /* Interpolate initial guess: */
            const auto nu =
                0.5 * std::tanh(10. * (x - 0.5 * (x_right + x_left)) /
                                (x_right - x_left));
            double v =
                velocity_left_ * (0.5 - nu) + velocity_right_ * (nu + 0.5);
 
            auto [f, df] = psi(x, v);
 
            while (std::abs(f) > norm) {
              const double v_next = v - f / df;
 
              /* Also break if we made no progress: */
              if (std::abs(v_next - v) <
                  tol * 0.5 * (velocity_right_ + velocity_left_)) {
                v = v_next;
                break;
              }
 
              if (v_next < velocity_right_)
                v = 0.5 * (velocity_right_ + v);
              else if (v_next > velocity_left_)
                v = 0.5 * (velocity_left_ + v);
              else
                v = v_next;
 
              const auto [new_f, new_df] = psi(x, v);
              f = new_f;
              df = new_df;
            }
 
            return v;
          }; /* find_velocity */
        });
      }
 
      state_type compute(const dealii::Point<dim> &point, Number t) final
      {
        const auto view = hyperbolic_system_.template view<dim, Number>();
 
        /* (7.2) */
        const double R_infty = (gamma_ + 1) / (gamma_ - 1);
 
        /* (7.3) */
        const double x = point[0] - velocity_ * t;
        const double v = find_velocity(x);
        Assert(v >= velocity_right_, dealii::ExcInternalError());
        Assert(v <= velocity_left_, dealii::ExcInternalError());
        const double rho = density_left_ * velocity_left_ / v;
        Assert(rho > 0., dealii::ExcInternalError());
        const double e = 1. / (2. * gamma_) *
                         (R_infty * velocity_left_ * velocity_right_ - v * v);
        Assert(e > 0., dealii::ExcInternalError());
 
        using state_type_1d = typename Description::
            template HyperbolicSystemView<1, Number>::state_type;
        const auto state_1d = state_type_1d{
            {Number(rho),
             Number(rho * (velocity_ + v)),
             Number(rho * (e + 0.5 * (velocity_ + v) * (velocity_ + v)))}};
 
        return view.expand_state(state_1d);
      }
 
    private:
      const HyperbolicSystem &hyperbolic_system_;
      Number gamma_;
 
      Number velocity_;
      Number velocity_left_;
      Number velocity_right_;
      Number density_left_;
      Number mu_;
      std::function<std::tuple<double, double>(double, double)> psi;
      std::function<double(double)> find_velocity;
    };
 
  } // namespace EulerInitialStates
} // namespace ryujin