limiter.h

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//
// SPDX-License-Identifier: Apache-2.0
// [LANL Copyright Statement]
// Copyright (C) 2023 - 2026 by the ryujin authors
// Copyright (C) 2023 - 2024 by Triad National Security, LLC
//
 
#pragma once
 
#include <ryujin/base/compile_time_options.h>
#include <ryujin/base/gpu.h>
#include <ryujin/base/newton.h>
#include <ryujin/base/observer_pointer.h>
#include <ryujin/base/simd.h>
#include <ryujin/linear_algebra/multicomponent_vector.h>
#include <ryujin/shallow_water/hyperbolic_system.h>
 
// #define DEBUG_OUTPUT_LIMITER
 
namespace ryujin
{
  namespace ShallowWater
  {
    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_("shallow_water_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_v<ScalarNumber, double>)
          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 PrecomputedVectorView = typename View::PrecomputedVectorView;
 
 
      static constexpr unsigned int n_bounds = 3;
 
      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_;
 
      flux_contribution_type flux_i_;
 
      Bounds bounds_;
 
      /* for relaxation */
 
      Number h_relaxation_numerator_;
      Number v2_relaxation_numerator_;
      Number relaxation_denominator_;
 
    };
 
 
    /*
     * -------------------------------------------------------------------------
     * 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 h_i = view_.water_depth(U_i);
      const auto v_i =
          view_.momentum(U_i) * view_.inverse_water_depth_mollified(U_i);
      const auto v2_i = v_i.norm_square();
 
      return {/*h_min*/ h_i, /*h_max*/ h_i, /*v2_max*/ v2_i};
    }
 
 
    template <int dim, typename Number, typename MemorySpace>
    DEAL_II_HOST_DEVICE_ALWAYS_INLINE auto
    LimiterView<dim, Number, MemorySpace>::combine_bounds(
        const Bounds &bounds_l, const Bounds &bounds_r) const -> Bounds
    {
      const auto &[h_min_l, h_max_l, v2_max_l] = bounds_l;
      const auto &[h_min_r, h_max_r, v2_max_r] = bounds_r;
 
      return {std::min(h_min_l, h_min_r),
              std::max(h_max_l, h_max_r),
              std::max(v2_max_l, v2_max_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
    {
      auto relaxed_bounds = bounds;
      auto &[h_min, h_max, v2_max] = 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();
      h_min *= std::max((Number(1.) - r), Number(eps));
      h_max *= (Number(1.) + r);
      v2_max *= (Number(1.) + r);
 
      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*/,
        unsigned int /*i*/,
        const state_type & /*U_i*/,
        const flux_contribution_type &flux_i)
    {
      flux_i_ = flux_i;
 
      auto &[h_min, h_max, v2_max] = bounds_;
 
      h_min = Number(std::numeric_limits<ScalarNumber>::max());
      h_max = Number(0.);
      v2_max = Number(0.);
 
      h_relaxation_numerator_ = Number(0.);
      v2_relaxation_numerator_ = Number(0.);
      relaxation_denominator_ = 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)
    {
      const auto &[U_i, Z_i, V_i] = flux_i_;
      const auto &[U_j, Z_j, V_j] = flux_j;
      const auto &[U_star_ij, U_star_ji] =
          view_.equilibrated_states(flux_i_, flux_j);
 
      /* bar state shifted by an affine shift: */
 
      const auto p_star_ij = view_.pressure(U_star_ij);
      const auto p_star_ji = view_.pressure(U_star_ji);
 
      auto U_ij_bar = U_star_ij * (Number(1.) + V_i * scaled_c_ij) +
                      U_star_ji * (Number(1.) - V_j * scaled_c_ij);
      for (unsigned int d = 0; d < dim; ++d)
        U_ij_bar[1 + d] += (p_star_ij - p_star_ji) * scaled_c_ij[d];
      U_ij_bar = ScalarNumber(0.5) * U_ij_bar + affine_shift;
 
      /* Bounds: */
 
      auto &[h_min, h_max, v2_max] = bounds_;
 
      const auto h_bar_ij = view_.water_depth(U_ij_bar);
      h_min = std::min(h_min, h_bar_ij);
      h_max = std::max(h_max, h_bar_ij);
 
      const auto v_bar_ij = view_.momentum(U_ij_bar) *
                            view_.inverse_water_depth_mollified(U_ij_bar);
      const auto v2_bar_ij = v_bar_ij.norm_square();
      v2_max = std::max(v2_max, v2_bar_ij);
 
      /* Relaxation: */
 
      /* Use a uniform weight. */
      const auto beta_ij = Number(1.);
 
      relaxation_denominator_ += std::abs(beta_ij);
 
      const auto h_i = view_.water_depth(U_i);
      const auto h_j = view_.water_depth(U_j);
      h_relaxation_numerator_ += beta_ij * (h_i + h_j);
 
      v2_relaxation_numerator_ +=
          beta_ij * (-V_i.norm_square() + V_j.norm_square());
    }
 
 
    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 &[h_min, h_max, v2_max] = bounds_;
 
      auto relaxed_bounds = fully_relax_bounds(bounds_, hd_i);
      auto &[h_min_relaxed, h_max_relaxed, v2_max_relaxed] = relaxed_bounds;
 
      /* Apply a stricter window: */
 
      constexpr ScalarNumber eps = std::numeric_limits<ScalarNumber>::epsilon();
 
      const Number h_relaxed = ScalarNumber(2. * relaxation_factor()) *
                               std::abs(h_relaxation_numerator_) /
                               (relaxation_denominator_ + Number(eps));
 
      const Number v2_relaxed = ScalarNumber(2. * relaxation_factor()) *
                                std::abs(v2_relaxation_numerator_) /
                                (relaxation_denominator_ + Number(eps));
 
      h_min_relaxed = std::max(h_min_relaxed, h_min - h_relaxed);
      h_max_relaxed = std::min(h_max_relaxed, h_max + h_relaxed);
      v2_max_relaxed = std::min(v2_max_relaxed, v2_max + v2_relaxed);
 
      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_l = t_min;
      Number t_r = t_max;
 
      const auto &[h_min, h_max, v2_max] = bounds;
 
      constexpr ScalarNumber min = std::numeric_limits<ScalarNumber>::min();
      constexpr ScalarNumber eps = std::numeric_limits<ScalarNumber>::epsilon();
      const auto small = view_.dry_state_relaxation_small();
      const auto large = view_.dry_state_relaxation_large();
      const auto relax_small = ScalarNumber(1. + small * eps);
      const auto relax = ScalarNumber(1. + large * eps);
 
      /*
       * We first limit the water_depth h.
       *
       * See [Guermond et al, 2021] (5.7).
       */
 
      {
        auto h_U = view_.water_depth(U);
        const auto &h_P = view_.water_depth(P);
 
        const auto test_min = view_.filter_dry_water_depth(
            std::max(Number(0.), h_U - relax * h_max));
        const auto test_max = view_.filter_dry_water_depth(
            std::max(Number(0.), h_min - relax * h_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 water depth (critical)!\n"
                    << "\n\t\th min:         " << h_min
                    << "\n\t\th min (delta): " << negative_part(h_U - h_min)
                    << "\n\t\th:             " << h_U
                    << "\n\t\th max (delta): " << positive_part(h_U - h_max)
                    << "\n\t\th max:         " << h_max << "\n"
                    << std::endl;
#endif
          success = false;
        }
 
        const Number denominator =
            ScalarNumber(1.) / (std::abs(h_P) + eps * h_max + min);
 
        constexpr auto lt = dealii::SIMDComparison::less_than;
 
        t_r = ryujin::compare_and_apply_mask<lt>( //
            h_max,
            h_U + t_r * h_P,
            /*
             * h_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.
             */
            (h_max - h_U) * denominator,
            t_r);
 
        t_r = ryujin::compare_and_apply_mask<lt>( //
            h_U + t_r * h_P,
            h_min,
            /*
             * h_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.
             */
            (h_U - h_min) * denominator,
            t_r);
 
        /*
         * Ensure that t_min <= t <= t_max. This might not be the case if
         * h_U is outside the interval [h_min, h_max]. Furthermore, the
         * quotient we take above is prone to numerical cancellation in
         * particular in the second pass of the limiter when h_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 h_new = view_.water_depth(U + t_r * P);
        const auto test_new_min = view_.filter_dry_water_depth(
            std::max(Number(0.), h_new - relax * h_max));
        const auto test_new_max = view_.filter_dry_water_depth(
            std::max(Number(0.), h_min - relax * h_new));
 
        if (!(test_new_min == Number(0.) && test_new_max == Number(0.))) {
#ifdef DEBUG_OUTPUT
          std::cout << std::fixed << std::setprecision(30);
          std::cout << "Bounds violation: high-order water depth!\n"
                    << "\n\t\th min:         " << h_min
                    << "\n\t\th min (delta): " << negative_part(h_new - h_min)
                    << "\n\t\th:             " << h_new
                    << "\n\t\th max (delta): " << positive_part(h_new - h_max)
                    << "\n\t\th max:         " << h_max << "\n"
                    << std::endl;
#endif
          success = false;
        }
#endif
      }
 
      /*
       * Limit the (negative) |v|^2:
       *
       * 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 the function:
       *
       *   psi = h^2 (|v|^2)^max - |q|^2
       */
 
      {
        /* We first check if t_r is a good state */
 
        const auto U_r = U + t_r * P;
        const auto h_r = view_.water_depth(U_r);
        const auto q_r = view_.momentum(U_r);
 
        const auto psi_r = relax_small * h_r * h_r * v2_max - q_r.norm_square();
 
        /*
         * 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 we can return: */
        if (t_l == t_r)
          return {t_l, success};
 
#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
 
        const auto U_l = U + t_l * P;
        const auto h_l = view_.water_depth(U_l);
        const auto q_l = view_.momentum(U_l);
 
        const auto psi_l = relax_small * h_l * h_l * v2_max - q_l.norm_square();
 
        /*
         * Verify that the left state is within bounds. This property might
         * be violated for relative CFL numbers larger than 1.
         *
         * We use a non-scaled eps here to force the lower_bound to be
         * negative so that we do not accidentally trigger in "perfect" dry
         * states with h_l equal to zero.
         */
        const auto filtered_h_l = view_.filter_dry_water_depth(h_l);
        const auto lower_bound =
            (ScalarNumber(1.) - relax) * filtered_h_l * filtered_h_l * v2_max -
            ScalarNumber(100.) * eps;
        if (!(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 square velocity (critical)!\n";
          std::cout << "\t\tPsi left: 0 <= " << psi_l << "\n" << std::endl;
#endif
          success = false;
        }
 
        /*
         * Skip the quadratic Newton step 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.))) {
          /*
           * If the bound is not satisfied, we need to find the root of a
           * quadratic function:
           *
           * psi(t)   = r (h_U + t h_P)^2 v2_max
           *            - (|q_U|^2 + 2(q_U * q_P) t + |q_P|^2 t^2)
           *
           * d_psi(t) = 2 r (h_U + t * h_P) * h_P v2_max
           *            - 2 (q_U * q_P) - 2 |q_P|^2 t
           *
           * where r = relax_small.
           *
           * We can compute the root of this function efficiently by using our
           * standard quadratic_newton_step() function that will use the points
           * [p1, p1, p2] as well as [p1, p2, p2] to construct two quadratic
           * polynomials to compute new candiates for the bounds [t_l, t_r]. In
           * case of a quadratic function psi(t) both polynomials will coincide
           * so that (up to round-off error) t_l = t_r.
           */
          const auto &h_U = view_.water_depth(U);
          const auto &h_P = view_.water_depth(P);
          const auto &q_U = view_.momentum(U);
          const auto &q_P = view_.momentum(P);
 
          const auto dpsi_l = ScalarNumber(2.) *
                              (relax_small * (h_U + t_l * h_P) * h_P * v2_max -
                               ((q_U * q_P) + q_P * q_P * t_l));
          const auto dpsi_r = ScalarNumber(2.) *
                              (relax_small * (h_U + t_r * h_P) * h_P * v2_max -
                               ((q_U * q_P) + q_P * q_P * t_r));
 
          quadratic_newton_step(
              t_l, t_r, psi_l, psi_r, dpsi_l, dpsi_r, Number(-1.));
 
#ifdef DEBUG_OUTPUT_LIMITER
          if (std::max(Number(0.), psi_r + Number(eps)) == Number(0.)) {
            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: (end)   " << t_l << std::endl;
            std::cout << "t_r: (end)   " << 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 h_new = view_.water_depth(U_new);
          const auto q_new = view_.momentum(U_new);
 
          const auto psi_new =
              relax_small * h_new * h_new * v2_max - q_new.norm_square();
 
          const auto lower_bound =
              (ScalarNumber(1.) - relax) * h_new * h_new * v2_max -
              ScalarNumber(100.) * eps;
 
          const bool psi_valid =
              std::min(Number(0.), psi_new - lower_bound) == Number(0.);
          if (!psi_valid) {
#ifdef DEBUG_OUTPUT
            std::cout << std::fixed << std::setprecision(16);
            std::cout << "Bounds violation: high-order square velocity!\n";
            std::cout << "\t\tPsi: 0 <= " << psi_new << "\n" << std::endl;
#endif
            success = false;
          }
        }
#endif
      }
 
      return {t_l, success};
    }
  } // namespace ShallowWater
} // namespace ryujin