limiter.template.h

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//
// SPDX-License-Identifier: MIT
// Copyright (C) 2020 - 2023 by the ryujin authors
//
 
#pragma once
 
#include "limiter.h"
 
namespace ryujin
{
  namespace ShallowWater
  {
    template <int dim, typename Number>
    std::tuple<Number, bool>
    Limiter<dim, Number>::limit(const Bounds &bounds,
                                const state_type &U,
                                const state_type &P,
                                const Number t_min /* = Number(0.) */,
                                const Number t_max /* = Number(1.) */)
    {
      bool success = true;
 
      Number t_l = t_min;
      Number t_r = t_max;
 
      constexpr ScalarNumber min = std::numeric_limits<ScalarNumber>::min();
      constexpr ScalarNumber eps = std::numeric_limits<ScalarNumber>::epsilon();
      const ScalarNumber relax_small = ScalarNumber(
          1. + hyperbolic_system.dry_state_relaxation_sharp() * eps); // FIXME
      const ScalarNumber relax =
          ScalarNumber(1. + hyperbolic_system.dry_state_relaxation_mollified() *
                                eps); // FIXME
 
      /*
       * We first limit the water_depth h.
       *
       * See [Guermond et al, 2021] (5.7).
       */
      {
        auto h_U = hyperbolic_system.water_depth(U);
        const auto &h_P = hyperbolic_system.water_depth(P);
        const auto &h_min = std::get<0>(bounds);
        const auto &h_max = std::get<1>(bounds);
 
        const auto test_min = hyperbolic_system.filter_dry_water_depth(
            std::max(Number(0.), h_U - relax * h_max));
        const auto test_max = hyperbolic_system.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";
          std::cout << "\t\th min: " << h_min << "\n";
          std::cout << "\t\th:     " << h_U << "\n";
          std::cout << "\t\th max: " << h_max << "\n" << std::endl;
#endif
          success = false;
        }
 
        const Number denominator =
            h_P / std::max(std::abs(h_P * h_P), Number(100. * min));
        constexpr auto lte = dealii::SIMDComparison::less_than_or_equal;
 
        t_r = dealii::compare_and_apply_mask<lte>(h_max,
                                                  h_U + t_r * h_P,
                                                  positive_part(h_max - h_U) *
                                                      denominator,
                                                  t_r);
 
        t_r = dealii::compare_and_apply_mask<lte>(h_U + t_r * h_P,
                                                  h_min,
                                                  positive_part(h_min - h_U) *
                                                      denominator,
                                                  t_r);
 
        /* Box back into bounds: */
        t_r = std::min(t_r, t_max);
        t_r = std::max(t_r, t_min);
 
        /*
         * Enforce strict limiting on dry states:
         */
        t_r = dealii::compare_and_apply_mask<dealii::SIMDComparison::equal>(
            hyperbolic_system.filter_dry_water_depth(h_U),
            Number(0.),
            t_l,
            t_r);
 
#ifdef CHECK_BOUNDS
        /*
         * Verify that the new state is within bounds:
         */
        const auto h_new = hyperbolic_system.water_depth(U + t_r * P);
        const auto test_new_min = hyperbolic_system.filter_dry_water_depth(
            std::max(Number(0.), h_new - relax * h_max));
        const auto test_new_max = hyperbolic_system.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";
          std::cout << "\t\th min: " << h_min << "\n";
          std::cout << "\t\th:     " << h_new << "\n";
          std::cout << "\t\th max: " << h_max << "\n";
          std::cout << "\t\th_U:   " << h_U << "\n";
          std::cout << "\t\th_P:   " << h_P << "\n";
          std::cout << "\t\tt_r:   " << t_r << "\n" << std::endl;
#endif
          success = false;
        }
#endif
      }
 
      /*
       * Then limit the (negative) kinetic energy:
       *
       * See [Guermond et al, 2021] (5.10).
       *
       * 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 KE_i^max - 1/2 |q|^2
       */
 
      {
        const auto &kin_max = std::get<2>(bounds);
 
        /* We first check if t_r is a good state */
 
        const auto U_r = U + t_r * P;
        const auto h_r = hyperbolic_system.water_depth(U_r);
        const auto q_r = hyperbolic_system.momentum(U_r);
 
        const auto psi_r =
            relax_small * h_r * kin_max - ScalarNumber(0.5) * 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 = dealii::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 = hyperbolic_system.water_depth(U_l);
        const auto q_l = hyperbolic_system.momentum(U_l);
 
        const auto psi_l =
            relax_small * h_l * kin_max - ScalarNumber(0.5) * 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 = hyperbolic_system.filter_dry_water_depth(h_l);
        const auto lower_bound =
            (ScalarNumber(1.) - relax) * filtered_h_l * kin_max - eps;
        if (!(std::min(Number(0.), psi_l - lower_bound + min) == Number(0.))) {
#ifdef DEBUG_OUTPUT
          std::cout << std::fixed << std::setprecision(16);
          std::cout
              << "Bounds violation: low-order kinetic energy (critical)!\n";
          std::cout << "\t\tPsi left: 0 <= " << psi_l << "\n" << std::endl;
#endif
          success = false;
        }
 
        /*
         * Return if the window between t_l and t_r is within the prescribed
         * tolerance:
         */
        if (std::max(Number(0.), t_r - t_l - newton_tolerance) == Number(0.))
          return {t_l, success};
 
        /*
         * If the bound is not satisfied, we need to find the root of a
         * quadratic function:
         *
         * psi(t)   = (h_U + t h_P) kin_max
         *            - 1/2 (|q_U|^2 + 2(q_U * q_P) t + |q_P|^2 t^2)
         *
         * d_psi(t) = h_P kin_max - (q_U * q_P) - |q_P|^2 t
         *
         * 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_P = hyperbolic_system.water_depth(P);
        const auto &q_U = hyperbolic_system.momentum(U);
        const auto &q_P = hyperbolic_system.momentum(P);
 
        const auto dpsi_l = h_P * kin_max - (q_U * q_P) - q_P * q_P * t_l;
        const auto dpsi_r = h_P * kin_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 CHECK_BOUNDS
        /*
         * Verify that the new state is within bounds:
         */
        {
          const auto U_new = U + t_l * P;
          const auto h_new = hyperbolic_system.water_depth(U_new);
          const auto q_new = hyperbolic_system.momentum(U_new);
 
          const auto psi_new = relax_small * h_new * kin_max -
                               ScalarNumber(0.5) * q_new.norm_square();
 
          const auto lower_bound =
              (ScalarNumber(1.) - relax) * h_new * kin_max - eps;
 
          const bool psi_valid =
              std::min(Number(0.), psi_new - lower_bound + min) == Number(0.);
          if (!psi_valid) {
#ifdef DEBUG_OUTPUT
            std::cout << std::fixed << std::setprecision(16);
            std::cout << "High-order bounds violation!\n";
            std::cout << "\t\tDepth   = " << h_new << "\n";
            std::cout << "\t\tPsi: 0 <= " << psi_new << "\n" << std::endl;
#endif
            success = false;
          }
        }
#endif
      }
 
      return {t_l, success};
    }
 
  } // namespace ShallowWater
} // namespace ryujin