initial_state_exact_riemann_solution.h

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//
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
// Copyright (C) 2023 - 2024 by the ryujin authors
// Copyright (C) 2025 by Triad National Security, LLC
//
 
#pragma once
 
#include <cmath>
#include <deal.II/base/tensor.h>
 
#include <initial_state_library.h>
 
// #define DEBUG_SOLUTION
 
namespace ryujin
{
  namespace EulerInitialStates
  {
    template <typename Description, int dim, typename Number>
    class ExactRiemannSolution : 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;
 
      using ScalarNumber = typename View::ScalarNumber;
 
 
      ExactRiemannSolution(const HyperbolicSystem &hyperbolic_system,
                           const std::string subsection)
          : InitialState<Description, dim, Number>("exact riemann solution",
                                                   subsection)
          , hyperbolic_system_(hyperbolic_system)
      {
        gamma_ = 1.4;
        if constexpr (!View::have_gamma) {
          this->add_parameter("gamma", gamma_, "The ratio of specific heats");
        }
 
        primitive_left_[0] = 1.4;
        primitive_left_[1] = 0.0;
        primitive_left_[2] = 1.0;
        this->add_parameter(
            "primitive state left",
            primitive_left_,
            "Initial 1d primitive state (rho, u, p) on the left");
 
        primitive_right_[0] = 1.4;
        primitive_right_[1] = 0.0;
        primitive_right_[2] = 1.0;
        this->add_parameter(
            "primitive state right",
            primitive_right_,
            "Initial 1d primitive state (rho, u, p) on the right");
 
        // Convert the primitive states to conserved states
        const auto prepare_riemann_data = [&]() {
          const auto view = hyperbolic_system_.template view<dim, Number>();
          if constexpr (View::have_gamma) {
            gamma_ = view.gamma();
          }
 
          const Number p_L = primitive_left_[2];
          const Number p_R = primitive_right_[2];
 
          p_star_ = compute_pstar(p_L, p_R, primitive_left_, primitive_right_);
 
          const Number u_L = primitive_left_[1];
          u_star_ = u_L - fZofP(p_star_, primitive_left_);
 
#ifdef DEBUG_SOLUTION
          const Number u_R = primitive_right_[1];
          std::cout << "left data          = " << primitive_left_
                    << "\nright data       = " << primitive_right_
                    << "\np_star           = " << p_star_
                    << "\nu_star           = " << u_star_
                    << "\nVerifying u_star = "
                    << u_R + fZofP(p_star_, primitive_right_) << std::endl;
#endif
 
          lambda_left_minus_ = lambda(p_star_, primitive_left_, -1.);
          lambda_left_plus_ =
              lambda_intermediate(p_star_, primitive_left_, -1.);
          lambda_right_minus_ =
              lambda_intermediate(p_star_, primitive_right_, 1.);
          lambda_right_plus_ = lambda(p_star_, primitive_right_, 1.);
 
 
#ifdef DEBUG_SOLUTION
          std::cout << "lambda_left_minus  =  " << lambda_left_minus_
                    << "\nlambda_left_plus   =  " << lambda_left_plus_
                    << "\nlambda_right_minus = " << lambda_right_minus_
                    << "\nlambda_right_plus  =  " << lambda_right_plus_
                    << std::endl;
#endif
        };
 
        this->parse_parameters_call_back.connect(prepare_riemann_data);
        prepare_riemann_data();
      }
 
 
      state_type compute(const dealii::Point<dim> &point, Number t) final
      {
        const auto view = hyperbolic_system_.template view<dim, Number>();
 
        const double &x = point[0];
 
        const Number xi = x / t;
 
        dealii::Tensor<1, 3, Number> primitive_state;
 
        if (t < 1.e-14 && x < 0.) {
          primitive_state = primitive_left_;
#ifdef DEBUG_SOLUTION
          std::cout << "Left primitive state: " << primitive_state << std::endl;
#endif
 
        } else if (t < 1.e-14 && x > 0.) {
          primitive_state = primitive_right_;
#ifdef DEBUG_SOLUTION
          std::cout << "Right primitive state: " << primitive_state
                    << std::endl;
#endif
 
        } else if (xi < lambda_left_minus_) {
          /* Left state: */
          primitive_state = primitive_left_;
#ifdef DEBUG_SOLUTION
          std::cout << "Left primitive state: " << primitive_state << std::endl;
#endif
 
        } else if (xi < lambda_left_plus_) {
          const auto c_LL =
              expansion_solution(p_star_, xi, primitive_left_, -1.);
          primitive_state = c_LL;
#ifdef DEBUG_SOLUTION
          std::cout << "Left expansion state: " << primitive_state << std::endl;
#endif
 
        } else if (xi < u_star_) {
          primitive_state = cstar_solution(p_star_, u_star_, primitive_left_);
 
          const Number p_L = primitive_left_[2];
          if (p_star_ < p_L)
            primitive_state = expansion_solution(
                p_star_, lambda_left_plus_, primitive_left_, -1.);
#ifdef DEBUG_SOLUTION
          std::cout << "Left cstar state: " << primitive_state << std::endl;
#endif
 
        } else if (xi < lambda_right_minus_) {
          primitive_state = cstar_solution(p_star_, u_star_, primitive_right_);
 
          const Number p_R = primitive_right_[2];
          if (p_star_ < p_R)
            primitive_state = expansion_solution(
                p_star_, lambda_right_minus_, primitive_right_, 1.);
#ifdef DEBUG_SOLUTION
          std::cout << "Right cstar state: " << primitive_state << std::endl;
#endif
 
        } else if (xi < lambda_right_plus_) {
          primitive_state =
              expansion_solution(p_star_, xi, primitive_right_, 1.);
#ifdef DEBUG_SOLUTION
          std::cout << "Right expansion state: " << primitive_state
                    << std::endl;
#endif
 
        } else {
          /* Right state: */
          primitive_state = primitive_right_;
#ifdef DEBUG_SOLUTION
          std::cout << "Right primitive state: " << primitive_state
                    << std::endl;
#endif
        }
 
        return view.from_initial_state(primitive_state);
      }
 
    private:
 
 
      Number gamma_;
 
      dealii::Tensor<1, 3, Number> primitive_left_;
      dealii::Tensor<1, 3, Number> primitive_right_;
 
 
 
      const HyperbolicSystem &hyperbolic_system_;
 
      Number p_star_;
      Number u_star_;
      Number lambda_left_minus_;
      Number lambda_left_plus_;
      Number lambda_right_minus_;
      Number lambda_right_plus_;
 
 
 
      Number fZofP(const Number &p_in,
                   const dealii::Tensor<1, 3, Number> &data_in) const
      {
        // Get left/right data
        const Number rho_Z = data_in[0];
        const Number p_Z = data_in[2];
 
        const Number c_Z = std::sqrt(gamma_ * p_Z / rho_Z);
 
        const Number A_Z = 2. / (gamma_ + 1.) / rho_Z;
        const Number B_Z = (gamma_ - 1.) / (gamma_ + 1.) * p_Z;
 
        const Number exp = 0.5 * (gamma_ - 1.) / gamma_;
        Number left_brach = 2. * c_Z / (gamma_ - 1.);
        left_brach *= (std::pow(p_in / p_Z, exp) - 1.);
 
        Number f_of_p = (p_in - p_Z) * std::sqrt(A_Z / (p_in + B_Z));
 
        if (p_in <= p_Z)
          f_of_p = left_brach;
 
        return f_of_p;
      }
 
 
      Number dfZofP(const Number &p_in,
                    const dealii::Tensor<1, 3, Number> &data_in) const
      {
        // Get left/right data
        const Number rho_Z = data_in[0];
        const Number p_Z = data_in[2];
 
        const Number c_Z = std::sqrt(gamma_ * p_Z / rho_Z);
 
        const Number A_Z = 2. / (gamma_ + 1.) / rho_Z;
        const Number B_Z = (gamma_ - 1.) / (gamma_ + 1.) * p_Z;
 
        Number exp = 0.5 * (gamma_ - 1.) / gamma_;
        Number left_brach = 2. * c_Z / (gamma_ - 1.) * exp;
        exp -= 1.;
 
        left_brach *= std::pow(p_in / p_Z, exp - 1.) / p_Z;
 
        Number right_branch = std::pow(A_Z / (p_in + B_Z), 1.5);
        right_branch *= (2. * B_Z + p_in + p_Z) / (2. * A_Z);
 
        Number df_of_p = right_branch;
 
        if (p_in <= p_Z)
          df_of_p = left_brach;
 
        return df_of_p;
      }
 
 
      Number dphi(const Number &p_in,
                  const dealii::Tensor<1, 3, Number> &data_left,
                  const dealii::Tensor<1, 3, Number> &data_right) const
      {
        return dfZofP(p_in, data_left) + dfZofP(p_in, data_right);
      }
 
 
      Number phi(const Number &p_in,
                 const dealii::Tensor<1, 3, Number> &data_left,
                 const dealii::Tensor<1, 3, Number> &data_right) const
      {
        const Number u_L = data_left[1];
        const Number u_R = data_right[1];
 
        return fZofP(p_in, data_right) + fZofP(p_in, data_left) + u_R - u_L;
      }
 
 
      Number lambda(const Number &p_in,
                    const dealii::Tensor<1, 3, Number> &data_in,
                    const Number &sign) const
      {
        // Get left/right data
        const Number rho_Z = data_in[0];
        const Number u_Z = data_in[1];
        const Number p_Z = data_in[2];
 
        const Number c_Z = std::sqrt(gamma_ * p_Z / rho_Z);
 
        const Number radicand =
            1. + 0.5 * (gamma_ + 1.) / gamma_ * std::max(p_in / p_Z - 1., 0.);
 
        return u_Z + sign * c_Z * std::sqrt(radicand);
      }
 
 
      Number lambda_intermediate(const Number &p_in,
                                 const dealii::Tensor<1, 3, Number> &data_in,
                                 const Number &sign) const
      {
        const Number rho_Z = data_in[0];
        const Number u_Z = data_in[1];
        const Number p_Z = data_in[2];
 
        const Number c_Z = std::sqrt(gamma_ * p_Z / rho_Z);
 
        const auto lambda_value = lambda(p_in, data_in, sign);
 
        const Number f_of_p = fZofP(p_in, data_in);
 
        const Number exp = 0.5 * (gamma_ - 1.) / gamma_;
        const Number expansion_speed =
            u_Z + sign * (f_of_p + c_Z * std::pow(p_in / p_Z, exp));
 
        Number result = lambda_value;
        if (p_in < p_Z)
          result = expansion_speed;
 
        return result;
      }
 
 
      dealii::Tensor<1, 3, Number>
      cstar_solution(const Number &p_star,
                     const Number &u_star,
                     const dealii::Tensor<1, 3, Number> &data_in) const
      {
        const Number rho_Z = data_in[0];
        const Number p_Z = data_in[2];
 
        // Define rho_star
        const Number p_ratio = p_star / p_Z;
        const Number gamma_ratio = (gamma_ - 1.) / (gamma_ + 1.);
 
        const Number numerator = rho_Z * (p_ratio + gamma_ratio);
        const Number denominator = gamma_ratio * p_ratio + 1.;
 
        Number rho_star = numerator / denominator;
 
        auto result = data_in;
        result[0] = rho_star;
        result[1] = u_star;
        result[2] = p_star;
 
        return result;
      }
 
 
      dealii::Tensor<1, 3, Number>
      expansion_solution(const Number & /*p_star*/,
                         const Number &xi,
                         const dealii::Tensor<1, 3, Number> &data_in,
                         const Number &sign) const
      {
        const Number rho_Z = data_in[0];
        const Number u_Z = data_in[1];
        const Number p_Z = data_in[2];
 
        const Number c_Z = std::sqrt(gamma_ * p_Z / rho_Z);
 
        // Define rho_expansion
        const Number gamma_ratio = (gamma_ - 1.) / (gamma_ + 1.);
 
        const Number first = 2. / (gamma_ + 1.);
        const Number second = gamma_ratio / c_Z * (u_Z - xi);
        const Number exp = 2. / (gamma_ - 1.);
 
        Number rho_expansion = rho_Z * std::pow(first - sign * second, exp);
 
        // Define p_expansion
        const Number factor = p_Z / std::pow(rho_Z, gamma_);
        const Number p_expansion = factor * std::pow(rho_expansion, gamma_);
 
        // Define u_expansion
        const Number u_expansion = u_Z + sign * fZofP(p_expansion, data_in);
 
        auto result = data_in;
        result[0] = rho_expansion;
        result[1] = u_expansion;
        result[2] = p_expansion;
 
        return result;
      }
 
 
      double compute_pstar(double p_1,
                           double p_2,
                           dealii::Tensor<1, 3, Number> data_1,
                           dealii::Tensor<1, 3, Number> data_2)
      {
        constexpr Number eps = std::numeric_limits<Number>::epsilon();
 
        // Ensure that p_1 <= p_2
 
        if (p_1 > p_2) {
          std::swap(p_1, p_2);
          std::swap(data_1, data_2);
        }
 
#ifdef DEBUG
        const double phi_1 = phi(p_1, data_1, data_2);
        const double phi_2 = phi(p_2, data_1, data_2);
        Assert(phi_1 * phi_2 <= 0.,
               dealii::ExcMessage(
                   "Euler::ExactRiemannSolver: failed to compute p_star."));
#endif
 
        //
        // We simply compute the root of phi with a bisection method down
        // to machine precision. This is not terribly efficient but luckily
        // happens only once during initialization.
        //
 
#ifdef DEBUG_SOLUTION
        std::cout << "Computing p_star with a bisection method." << std::endl;
#endif
 
        unsigned int iter = 0;
        for (; iter < 200; ++iter) {
 
          // Check for convergence:
          if (std::abs(p_2 - p_1) < 10. * eps * std::max(p_1, p_2)) {
            break;
          }
 
          const double phi_2 = phi(p_2, data_1, data_2);
 
#ifdef DEBUG_SOLUTION
          const double phi_1 = phi(p_1, data_1, data_2);
 
          std::cout << "\niter: " << iter << "\n";
          std::cout << "p_1: " << p_1 << "\n";
          std::cout << "p_2: " << p_2 << "\n";
          std::cout << "phi_1: " << phi_1 << "\n";
          std::cout << "phi_2: " << phi_2 << "\n";
#endif
 
          const auto p_m = 0.5 * (p_2 + p_1);
          const double phi_m = phi(p_m, data_1, data_2);
 
          if (phi_m * phi_2 >= 0.) {
            p_2 = p_m;
          } else {
            p_1 = p_m;
          }
        }
 
#ifdef DEBUG_SOLUTION
        const double phi_2 = phi(p_2, data_1, data_2);
        std::cout << "After " << iter << " iterations:"
                  << "\np_star =      " << p_2 << "\nphi(p_star) = " << phi_2
                  << "\n|p_2 - p_1| = " << std::abs(p_2 - p_1) << std::endl;
#endif
 
        return p_2;
      }
 
    };
  } // namespace EulerInitialStates
} // namespace ryujin