limiter.h

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//
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
// Copyright (C) 2020 - 2026 by the ryujin authors
//
 
#pragma once
 
#include <compile_time_options.h>
 
#include "hyperbolic_system.h"
 
#include <gpu.h>
#include <multicomponent_vector.h>
#include <newton.h>
#include <observer_pointer.h>
#include <simd.h>
 
// #define DEBUG_OUTPUT_LIMITER
 
namespace ryujin
{
  namespace EulerAEOS
  {
    template <int dim,
              typename Number = double,
              typename MemorySpace = dealii::MemorySpace::Host>
    class LimiterView;
 
    template <typename ScalarNumber = double>
    class Limiter : public dealii::ParameterAcceptor
    {
    public:
 
      struct Parameters {
        unsigned int iterations;
        double newton_tolerance;
        unsigned int newton_max_iterations;
        double relaxation_factor;
      };
 
 
 
      Limiter(const HyperbolicSystem &hyperbolic_system,
              const std::string &subsection = "/Limiter")
          : ParameterAcceptor(subsection)
          , parameters_("euler_aeos_limiter_parameters",
                        TransferPolicy::implicit_transfers_host_resident)
          , hyperbolic_system_(&hyperbolic_system)
      {
        /* reference remains valid due to implicit_transfers_host_resident */
        auto &parameters = *parameters_.view();
 
        parameters.iterations = 2;
        add_parameter("iterations",
                      parameters.iterations,
                      "Number of limiter iterations");
 
        if constexpr (std::is_same<ScalarNumber, double>::value)
          parameters.newton_tolerance = 1.e-10;
        else
          parameters.newton_tolerance = 1.e-4;
        add_parameter("newton tolerance",
                      parameters.newton_tolerance,
                      "Tolerance for the quadratic newton stopping criterion");
 
        parameters.newton_max_iterations = 2;
        add_parameter("newton max iterations",
                      parameters.newton_max_iterations,
                      "Maximal number of quadratic newton iterations performed "
                      "during limiting");
 
        parameters.relaxation_factor = 1.;
        add_parameter("relaxation factor",
                      parameters.relaxation_factor,
                      "Factor for scaling the relaxation window with r_i = "
                      "factor * (m_i/|Omega|)^(1.5/d).");
 
        /* invalidates view on default memory space */
        ParameterAcceptor::parse_parameters_call_back.connect(
            [this] { parameters_.view(); });
      }
 
      template <int dim,
                typename Number,
                typename MemorySpace = dealii::MemorySpace::Host>
      auto view() const
      {
        return LimiterView<dim, Number, MemorySpace>{
            hyperbolic_system_->template view<dim, Number, MemorySpace>(),
            *this};
      }
 
    private:
 
 
      Mirrored<Parameters> parameters_;
 
 
 
      dealii::ObserverPointer<const HyperbolicSystem> hyperbolic_system_;
 
      template <int, typename, typename>
      friend class LimiterView;
 
    };
 
 
    template <int dim, typename Number, typename MemorySpace>
    class LimiterView
    {
    public:
      static_assert(
          std::is_same_v<MemorySpace, dealii::MemorySpace::Host> ||
              std::is_same_v<MemorySpace, dealii::MemorySpace::Default>,
          "Unexpected memory space");
 
 
      using View = HyperbolicSystemView<dim, Number, MemorySpace>;
 
      using ScalarNumber = typename View::ScalarNumber;
 
      using state_type = typename View::state_type;
 
      using flux_contribution_type = typename View::flux_contribution_type;
 
      using precomputed_type = typename View::precomputed_type;
 
      using PrecomputedVectorView = typename View::PrecomputedVectorView;
 
 
      static constexpr unsigned int n_bounds = 4;
 
      using Bounds = std::array<Number, n_bounds>;
 
      LimiterView(const View &view, const Limiter<ScalarNumber> &limiter)
          : view_(view)
          , parameters_(limiter.parameters_.template view<MemorySpace>())
      {
      }
 
      DEAL_II_HOST_DEVICE_ALWAYS_INLINE unsigned int iterations() const
      {
        return parameters_->iterations;
      }
 
      DEAL_II_HOST_DEVICE_ALWAYS_INLINE ScalarNumber newton_tolerance() const
      {
        return ScalarNumber(parameters_->newton_tolerance);
      }
 
      DEAL_II_HOST_DEVICE_ALWAYS_INLINE unsigned int
      newton_max_iterations() const
      {
        return parameters_->newton_max_iterations;
      }
 
      DEAL_II_HOST_DEVICE_ALWAYS_INLINE ScalarNumber relaxation_factor() const
      {
        return ScalarNumber(parameters_->relaxation_factor);
      }
 
      DEAL_II_HOST_DEVICE Bounds
      projection_bounds_from_state(const PrecomputedVectorView &pv,
                                   const unsigned int i,
                                   const state_type &U_i) const;
 
      DEAL_II_HOST_DEVICE Bounds combine_bounds(
          const Bounds &bounds_left, const Bounds &bounds_right) const;
 
      DEAL_II_HOST_DEVICE Bounds fully_relax_bounds(const Bounds &bounds,
                                                    const Number &hd) const;
 
 
 
      DEAL_II_HOST_DEVICE void reset(const PrecomputedVectorView &pv,
                                     const unsigned int i,
                                     const state_type &U_i,
                                     const flux_contribution_type &flux_i);
 
      DEAL_II_HOST_DEVICE void
      accumulate(const PrecomputedVectorView &pv,
                 const unsigned int *js,
                 const state_type &U_j,
                 const flux_contribution_type &flux_j,
                 const dealii::Tensor<1, dim, Number> &scaled_c_ij,
                 const state_type &affine_shift);
 
      DEAL_II_HOST_DEVICE Bounds bounds(const Number hd_i) const;
 
 
 
      DEAL_II_HOST_DEVICE std::tuple<Number, bool>
      limit(const Bounds &bounds,
            const state_type &U,
            const state_type &P,
            const Number t_min = Number(0.),
            const Number t_max = Number(1.)) const;
 
    private:
 
 
      const View view_;
      const Limiter<ScalarNumber>::Parameters *const parameters_;
 
      state_type U_i_;
      flux_contribution_type flux_i_;
 
      Bounds bounds_;
 
      Number rho_relaxation_numerator_;
      Number rho_relaxation_denominator_;
      Number s_interp_max_;
 
    };
 
 
    /*
     * -------------------------------------------------------------------------
     * Inline definitions
     * -------------------------------------------------------------------------
     */
 
 
    template <int dim, typename Number, typename MemorySpace>
    DEAL_II_HOST_DEVICE_ALWAYS_INLINE auto
    LimiterView<dim, Number, MemorySpace>::projection_bounds_from_state(
        const PrecomputedVectorView &pv,
        const unsigned int i,
        const state_type &U_i) const -> Bounds
    {
      const auto rho_i = view_.density(U_i);
      const auto &[p_i, gamma_min_i, s_i, eta_i] =
          pv.template read_tensor<Number, precomputed_type>(i);
 
      return {/*rho_min*/ rho_i,
              /*rho_max*/ rho_i,
              /*s_min*/ s_i,
              /*gamma_min*/ gamma_min_i};
    }
 
 
    template <int dim, typename Number, typename MemorySpace>
    DEAL_II_HOST_DEVICE_ALWAYS_INLINE auto
    LimiterView<dim, Number, MemorySpace>::combine_bounds(
        const Bounds &bounds_left, const Bounds &bounds_right) const -> Bounds
    {
      const auto &[rho_min_l, rho_max_l, s_min_l, gamma_min_l] = bounds_left;
      const auto &[rho_min_r, rho_max_r, s_min_r, gamma_min_r] = bounds_right;
 
      return {std::min(rho_min_l, rho_min_r),
              std::max(rho_max_l, rho_max_r),
              std::min(s_min_l, s_min_r),
              std::min(gamma_min_l, gamma_min_r)};
    }
 
 
    template <int dim, typename Number, typename MemorySpace>
    DEAL_II_HOST_DEVICE_ALWAYS_INLINE auto
    LimiterView<dim, Number, MemorySpace>::fully_relax_bounds(
        const Bounds &bounds, const Number &hd) const -> Bounds
    {
      const auto &[rho_min, rho_max, s_min, gamma_min] = bounds;
 
      auto relaxed_bounds = bounds;
      auto &[rho_min_relaxed, rho_max_relaxed, s_min_relaxed, g_m_relaxed] =
          relaxed_bounds;
 
      /* Use r = factor * (m_i / |Omega|) ^ (1.5 / d): */
 
      Number r = std::sqrt(hd);                              // in 3D: ^ 3/6
      if constexpr (dim == 2)                                //
        r = dealii::Utilities::fixed_power<3>(std::sqrt(r)); // in 2D: ^ 3/4
      else if constexpr (dim == 1)                           //
        r = dealii::Utilities::fixed_power<3>(r);            // in 1D: ^ 3/2
      r *= relaxation_factor();
 
      constexpr ScalarNumber eps = std::numeric_limits<ScalarNumber>::epsilon();
      rho_min_relaxed *= std::max(Number(1.) - r, Number(eps));
      rho_max_relaxed *= (Number(1.) + r);
      s_min_relaxed *= std::max(Number(1.) - r, Number(eps));
 
      /*
       * If we have a maximum compressibility constant, b, the maximum
       * bound for rho changes. See @cite ryujin-2023-4 for how to define
       * rho_max.
       */
 
      const auto numerator = (gamma_min + Number(1.)) * rho_max;
      const auto covolume_b = view_.eos_covolume_constant();
      const auto denominator =
          gamma_min - Number(1.) + ScalarNumber(2.) * covolume_b * rho_max;
      const auto rho_compressibility_bound = numerator / denominator;
 
      rho_max_relaxed = std::min(rho_compressibility_bound, rho_max_relaxed);
 
      return relaxed_bounds;
    }
 
 
    template <int dim, typename Number, typename MemorySpace>
    DEAL_II_HOST_DEVICE_ALWAYS_INLINE void
    LimiterView<dim, Number, MemorySpace>::reset(
        const PrecomputedVectorView &pv,
        const unsigned int i,
        const state_type &U_i,
        const flux_contribution_type &flux_i)
    {
      U_i_ = U_i;
      flux_i_ = flux_i;
 
      /* Bounds: */
 
      auto &[rho_min, rho_max, s_min, gamma_min] = bounds_;
 
      rho_min = Number(std::numeric_limits<ScalarNumber>::max());
      rho_max = Number(0.);
      s_min = Number(std::numeric_limits<ScalarNumber>::max());
 
      const auto &[p_i, gamma_min_i, s_i, eta_i] =
          pv.template read_tensor<Number, precomputed_type>(i);
 
      gamma_min = gamma_min_i;
 
      /* Relaxation: */
 
      rho_relaxation_numerator_ = Number(0.);
      rho_relaxation_denominator_ = Number(0.);
      s_interp_max_ = Number(0.);
    }
 
 
    template <int dim, typename Number, typename MemorySpace>
    DEAL_II_HOST_DEVICE_ALWAYS_INLINE void
    LimiterView<dim, Number, MemorySpace>::accumulate(
        const PrecomputedVectorView &pv,
        const unsigned int *js,
        const state_type &U_j,
        const flux_contribution_type &flux_j,
        const dealii::Tensor<1, dim, Number> &scaled_c_ij,
        const state_type &affine_shift)
    {
      // TODO: Currently we only apply the affine_shift to U_ij_bar (which
      // then enters all bounds), but we do not modify s_interp and
      // rho_relaxation. When actually adding a source term to the Euler
      // equations verify that this does the right thing.
      Assert(std::max(affine_shift.norm(), Number(0.)) == Number(0.),
             dealii::ExcNotImplemented());
 
      /* Bounds: */
      auto &[rho_min, rho_max, s_min, gamma_min] = bounds_;
 
      const auto rho_i = view_.density(U_i_);
      const auto rho_j = view_.density(U_j);
 
      /* bar state shifted by an affine shift: */
      const auto U_ij_bar =
          ScalarNumber(0.5) * (U_i_ + U_j) -
          ScalarNumber(0.5) * contract(add(flux_j, -flux_i_), scaled_c_ij) +
          affine_shift;
 
      const auto rho_ij_bar = view_.density(U_ij_bar);
 
      /* Density bounds: */
 
      rho_min = std::min(rho_min, rho_ij_bar);
      rho_max = std::max(rho_max, rho_ij_bar);
 
      /* Density relaxation: */
 
      /* Use a uniform weight. */
      const auto beta_ij = Number(1.);
      rho_relaxation_numerator_ += beta_ij * (rho_i + rho_j);
      rho_relaxation_denominator_ += std::abs(beta_ij);
 
      /* Surrogate entropy bounds and relaxation: */
 
      if (view_.compute_strict_bounds()) {
        /*
         * Compute strict bounds precisely as outlined in @cite ryujin-2023-4
         *
         * This means, we compute
         *  - the surrogate entropy at dof j with the gamma_min of index i,
         *  - the surrogate entropy of the bar state U_ij_bar
         *  - an interpolated surrogate entropy at (U_i + U_j) / 2 for
         *    bounds relaxation:
         */
 
        const auto s_j = view_.surrogate_specific_entropy(U_j, gamma_min);
 
        const auto s_ij_bar =
            view_.surrogate_specific_entropy(U_ij_bar, gamma_min);
 
        const Number s_interp = view_.surrogate_specific_entropy(
            (U_i_ + U_j) * ScalarNumber(.5), gamma_min);
 
        s_min = std::min(s_min, s_j);
        s_min = std::min(s_min, s_ij_bar);
        s_interp_max_ = std::max(s_interp_max_, s_interp);
 
      } else {
        /*
         * Compute a cheaper bound solely relying on the diagonal s_j
         * (computed with gamma_min_j) and the surrogate entropy s_ij_bar
         * of the bar state. We use the s_ij_bar for computing the bounds
         * relaxation as well.
         */
        const auto [p_j, gamma_min_j, s_j, eta_j] =
            pv.template read_tensor<Number, precomputed_type>(js);
 
        const auto s_ij_bar =
            view_.surrogate_specific_entropy(U_ij_bar, gamma_min);
 
        s_min = std::min(s_min, s_j);
        s_min = std::min(s_min, s_ij_bar);
        s_interp_max_ = std::max(s_interp_max_, s_ij_bar);
      }
    }
 
 
    template <int dim, typename Number, typename MemorySpace>
    DEAL_II_HOST_DEVICE_ALWAYS_INLINE auto
    LimiterView<dim, Number, MemorySpace>::bounds(const Number hd_i) const
        -> Bounds
    {
      const auto &[rho_min, rho_max, s_min, gamma_min] = bounds_;
 
      auto relaxed_bounds = fully_relax_bounds(bounds_, hd_i);
      auto &[rho_min_relaxed, rho_max_relaxed, s_min_relaxed, g_m_relaxed] =
          relaxed_bounds;
 
      /* Apply a stricter window: */
 
      constexpr ScalarNumber eps = std::numeric_limits<ScalarNumber>::epsilon();
 
      const auto rho_relaxation =
          ScalarNumber(2. * relaxation_factor()) *
          std::abs(rho_relaxation_numerator_) /
          (std::abs(rho_relaxation_denominator_) + Number(eps));
 
      const auto entropy_relaxation =
          relaxation_factor() * (s_interp_max_ - s_min);
 
      rho_min_relaxed = std::max(rho_min_relaxed, rho_min - rho_relaxation);
      rho_max_relaxed = std::min(rho_max_relaxed, rho_max + rho_relaxation);
      s_min_relaxed = std::max(s_min_relaxed, s_min - entropy_relaxation);
 
      return relaxed_bounds;
    }
 
 
    template <int dim, typename Number, typename MemorySpace>
    DEAL_II_HOST_DEVICE_ALWAYS_INLINE std::tuple<Number, bool>
    LimiterView<dim, Number, MemorySpace>::limit(
        const Bounds &bounds,
        const state_type &U,
        const state_type &P,
        const Number t_min /* = Number(0.) */,
        const Number t_max /* = Number(1.) */) const
    {
      bool success = true;
      Number t_r = t_max;
 
      constexpr ScalarNumber eps = std::numeric_limits<ScalarNumber>::epsilon();
      const auto small = view_.vacuum_state_relaxation_small();
      const auto large = view_.vacuum_state_relaxation_large();
      const ScalarNumber relax_small = ScalarNumber(1. + small * eps);
      const ScalarNumber relax = ScalarNumber(1. + large * eps);
 
      /*
       * First limit the density rho.
       *
       * See [Guermond, Nazarov, Popov, Thomas] (4.8):
       */
 
      {
        const auto &rho_U = view_.density(U);
        const auto &rho_P = view_.density(P);
 
        const auto &rho_min = std::get<0>(bounds);
        const auto &rho_max = std::get<1>(bounds);
 
        /*
         * Verify that rho_U is within bounds. This property might be
         * violated for relative CFL numbers larger than 1.
         */
        const auto test_min = view_.filter_vacuum_density(
            std::max(Number(0.), rho_U - relax * rho_max));
        const auto test_max = view_.filter_vacuum_density(
            std::max(Number(0.), rho_min - relax * rho_U));
        if (!(test_min == Number(0.) && test_max == Number(0.))) {
#ifdef DEBUG_OUTPUT
          std::cout << std::fixed << std::setprecision(16);
          std::cout << "Bounds violation: low-order density (critical)!"
                    << "\n\t\trho min:         " << rho_min
                    << "\n\t\trho min (delta): "
                    << negative_part(rho_U - rho_min)
                    << "\n\t\trho:             " << rho_U
                    << "\n\t\trho max (delta): "
                    << positive_part(rho_U - rho_max)
                    << "\n\t\trho max:         " << rho_max << "\n"
                    << std::endl;
#endif
          success = false;
        }
 
        const Number denominator =
            ScalarNumber(1.) / (std::abs(rho_P) + eps * rho_max);
 
        t_r = ryujin::compare_and_apply_mask<dealii::SIMDComparison::less_than>(
            rho_max,
            rho_U + t_r * rho_P,
            /*
             * rho_P is positive.
             *
             * Note: Do not take an absolute value here. If we are out of
             * bounds we have to ensure that t_r is set to t_min.
             */
            (rho_max - rho_U) * denominator,
            t_r);
 
        t_r = ryujin::compare_and_apply_mask<dealii::SIMDComparison::less_than>(
            rho_U + t_r * rho_P,
            rho_min,
            /*
             * rho_P is negative.
             *
             * Note: Do not take an absolute value here. If we are out of
             * bounds we have to ensure that t_r is set to t_min.
             */
            (rho_U - rho_min) * denominator,
            t_r);
 
        /*
         * Ensure that t_min <= t <= t_max. This might not be the case if
         * rho_U is outside the interval [rho_min, rho_max]. Furthermore,
         * the quotient we take above is prone to numerical cancellation in
         * particular in the second pass of the limiter when rho_P might be
         * small.
         */
        t_r = std::min(t_r, t_max);
        t_r = std::max(t_r, t_min);
 
#ifdef DEBUG_EXPENSIVE_BOUNDS_CHECK
        /*
         * Verify that the new state is within bounds:
         */
        const auto rho_new = view_.density(U + t_r * P);
        const auto test_new_min = view_.filter_vacuum_density(
            std::max(Number(0.), rho_new - relax * rho_max));
        const auto test_new_max = view_.filter_vacuum_density(
            std::max(Number(0.), rho_min - relax * rho_new));
        if (!(test_new_min == Number(0.) && test_new_max == Number(0.))) {
#ifdef DEBUG_OUTPUT
          std::cout << std::fixed << std::setprecision(16);
          std::cout << "Bounds violation: high-order density!"
                    << "\n\t\trho min:         " << rho_min
                    << "\n\t\trho min (delta): "
                    << negative_part(rho_new - rho_min)
                    << "\n\t\trho:             " << rho_new
                    << "\n\t\trho max (delta): "
                    << positive_part(rho_new - rho_max)
                    << "\n\t\trho max:         " << rho_max << "\n"
                    << std::endl;
#endif
          success = false;
        }
#endif
      }
 
      /*
       * Then limit the specific entropy:
       *
       * See [Guermond, Nazarov, Popov, Thomas],
       * Section 4.6 + Section 5.1
       * and @cite clayton2023robust Section 6:
       */
 
      Number t_l = t_min; // good state
 
      const auto &gamma = std::get<3>(bounds) /* = gamma_min*/;
      const Number gm1 = gamma - Number(1.);
 
      const auto b = Number(view_.eos_covolume_constant());
      const auto pinf = Number(view_.eos_interpolation_pinfty());
      const auto q = Number(view_.eos_interpolation_q());
 
      {
        /*
         * Prepare a quadratic Newton method:
         *
         * Given initial limiter values t_l and t_r with psi(t_l) > 0 and
         * psi(t_r) < 0 we try to find t^\ast with psi(t^\ast) \approx 0.
         *
         * Here, psi is a 3-convex function obtained by scaling the specific
         * entropy s:
         *
         *   psi = \rho ^ {\gamma + 1} s
         *
         * (s in turn was defined as s =\varepsilon \rho ^{-\gamma}, where
         * \varepsilon = (\rho e - pinf * (1 - b rho)) is the shifted
         * internal energy.)
         */
 
        const auto &s_min = std::get<2>(bounds);
 
#ifdef DEBUG_OUTPUT_LIMITER
        std::cout << std::endl;
        std::cout << std::fixed << std::setprecision(16);
        std::cout << "t_l: (start) " << t_l << std::endl;
        std::cout << "t_r: (start) " << t_r << std::endl;
#endif
 
        for (unsigned int n = 0; n < newton_max_iterations(); ++n) {
 
          const auto U_r = U + t_r * P;
          const auto rho_r = view_.density(U_r);
          const auto rho_r_gamma = ryujin::pow(rho_r, gamma);
          const auto covolume_r = Number(1.) - b * rho_r;
 
          const auto rho_e_r = view_.internal_energy(U_r);
          const auto shift_r = rho_e_r - rho_r * q - pinf * covolume_r;
 
          auto psi_r =
              relax_small * rho_r * shift_r -
              s_min * rho_r * rho_r_gamma * ryujin::pow(covolume_r, -gm1);
 
#ifndef DEBUG_EXPENSIVE_BOUNDS_CHECK
          /*
           * If psi_r > 0 the right state is fine, force returning t_r by
           * setting t_l = t_r:
           */
          t_l = ryujin::compare_and_apply_mask<
              dealii::SIMDComparison::greater_than>(
              psi_r, Number(0.), t_r, t_l);
 
          /*
           * If we have set t_l = t_r everywhere then all states state U_r
           * with t_r obey the specific entropy inequality and we can
           * break.
           *
           * This is a very important optimization: Only for 1 in (25 to
           * 50) cases do we actually need to limit on the specific entropy
           * because one of the right states failed. So we can skip
           * constructing the left state U_l, which is expensive.
           *
           * This implies unfortunately that we might not accurately report
           * whether the low_order update U itself obeyed bounds because
           * U_r = U + t_r * P pushed us back into bounds. We thus skip
           * this shortcut if `DEBUG_EXPENSIVE_BOUNDS_CHECK` is set.
           */
          if (t_l == t_r) {
#ifdef DEBUG_OUTPUT_LIMITER
            std::cout << "shortcut: t_l == t_r" << std::endl;
            std::cout << "psi_l:       " << psi_l << std::endl;
            std::cout << "psi_r:       " << psi_r << std::endl;
            std::cout << "t_l: (  " << n << "  ) " << t_l << std::endl;
            std::cout << "t_r: (  " << n << "  ) " << t_r << std::endl;
#endif
            break;
          }
#endif
 
          const auto U_l = U + t_l * P;
          const auto rho_l = view_.density(U_l);
          const auto rho_l_gamma = ryujin::pow(rho_l, gamma);
          const auto covolume_l = Number(1.) - b * rho_l;
          const auto rho_e_l = view_.internal_energy(U_l);
          const auto shift_l = rho_e_l - rho_l * q - pinf * covolume_l;
 
          auto psi_l =
              relax_small * rho_l * shift_l -
              s_min * rho_l * rho_l_gamma * ryujin::pow(covolume_l, -gm1);
 
          /*
           * Verify that the left state is within bounds. This property might
           * be violated for relative CFL numbers larger than 1.
           */
          const auto lower_bound = (ScalarNumber(1.) - relax) * s_min * rho_l *
                                   rho_l_gamma * ryujin::pow(covolume_l, -gm1);
          if (n == 0 &&
              !(std::min(Number(0.), psi_l - lower_bound) == Number(0.))) {
#ifdef DEBUG_OUTPUT
            std::cout << std::fixed << std::setprecision(16);
            std::cout
                << "Bounds violation: low-order specific entropy (critical)!\n";
            std::cout << "\t\tPsi left: 0 <= " << psi_l << "\n" << std::endl;
#endif
            success = false;
          }
 
#ifdef DEBUG_EXPENSIVE_BOUNDS_CHECK
          /*
           * If psi_r > 0 the right state is fine, force returning t_r by
           * setting t_l = t_r:
           */
          t_l = ryujin::compare_and_apply_mask<
              dealii::SIMDComparison::greater_than>(
              psi_r, Number(0.), t_r, t_l);
#endif
 
          /*
           * Break if the window between t_l and t_r is within the prescribed
           * tolerance:
           */
          const Number tolerance(newton_tolerance());
          if (std::max(Number(0.), t_r - t_l - tolerance) == Number(0.)) {
#ifdef DEBUG_OUTPUT_LIMITER
            std::cout << "break: t_l and t_r within tolerance" << std::endl;
            std::cout << "psi_l:       " << psi_l << std::endl;
            std::cout << "psi_r:       " << psi_r << std::endl;
            std::cout << "t_l: (  " << n << "  ) " << t_l << std::endl;
            std::cout << "t_r: (  " << n << "  ) " << t_r << std::endl;
#endif
            break;
          }
 
          /* We got unlucky and have to perform a Newton step: */
 
          const auto drho = view_.density(P);
          const auto drho_e_l = view_.internal_energy_derivative(U_l) * P;
          const auto drho_e_r = view_.internal_energy_derivative(U_r) * P;
 
          const auto q_pinf_term_l =
              ScalarNumber(2.) * rho_l * q +
              pinf * (Number(1.) - ScalarNumber(2.) * b * rho_l);
          const auto q_pinf_term_r =
              ScalarNumber(2.) * rho_r * q +
              pinf * (Number(1.) - ScalarNumber(2.) * b * rho_r);
 
          const auto extra_term_l = s_min *
                                    ryujin::pow(rho_l / covolume_l, gamma) *
                                    (covolume_l + gamma - b * rho_l);
          const auto extra_term_r = s_min *
                                    ryujin::pow(rho_r / covolume_r, gamma) *
                                    (covolume_r + gamma - b * rho_r);
 
          const auto dpsi_l = rho_l * drho_e_l +
                              (rho_e_l - q_pinf_term_l - extra_term_l) * drho;
          const auto dpsi_r = rho_r * drho_e_r +
                              (rho_e_r - q_pinf_term_r - extra_term_r) * drho;
 
          quadratic_newton_step(
              t_l, t_r, psi_l, psi_r, dpsi_l, dpsi_r, Number(-1.));
 
#ifdef DEBUG_OUTPUT_LIMITER
          std::cout << "psi_l:       " << psi_l << std::endl;
          std::cout << "psi_r:       " << psi_r << std::endl;
          std::cout << "dpsi_l:      " << dpsi_l << std::endl;
          std::cout << "dpsi_r:      " << dpsi_r << std::endl;
          std::cout << "t_l: (  " << n << "  ) " << t_l << std::endl;
          std::cout << "t_r: (  " << n << "  ) " << t_r << std::endl;
#endif
        }
 
#ifdef DEBUG_EXPENSIVE_BOUNDS_CHECK
        /*
         * Verify that the new state is within bounds:
         */
        {
          const auto U_new = U + t_l * P;
 
          const auto rho_new = view_.density(U_new);
          const auto covolume_new = Number(1.) - b * rho_new;
 
          const auto rho_new_gamma = ryujin::pow(rho_new, gamma);
          const auto rho_e_new = view_.internal_energy(U_new);
 
          const auto shift_new = rho_e_new - rho_new * q - pinf * covolume_new;
 
          const auto psi_new =
              relax_small * rho_new * shift_new -
              s_min * rho_new * rho_new_gamma * ryujin::pow(covolume_new, -gm1);
 
          const auto lower_bound = (ScalarNumber(1.) - relax) * s_min *
                                   rho_new * rho_new_gamma *
                                   ryujin::pow(covolume_new, -gm1);
 
          const bool e_valid = std::min(Number(0.), shift_new) == Number(0.);
          const bool psi_valid =
              std::min(Number(0.), psi_new - lower_bound) == Number(0.);
 
          if (!e_valid || !psi_valid) {
#ifdef DEBUG_OUTPUT
            std::cout << std::fixed << std::setprecision(16);
            std::cout << "Bounds violation: high-order specific entropy!\n";
            std::cout << "\t\trho e: 0 <= " << rho_e_new << "\n";
            std::cout << "\t\tPsi:   0 <= " << psi_new << "\n" << std::endl;
#endif
            success = false;
          }
        }
#endif
      }
 
      return {t_l, success};
    }
  } // namespace EulerAEOS
} // namespace ryujin