Line data Source code
1 0 : // Distributed under the MIT License.
2 : // See LICENSE.txt for details.
3 :
4 : #pragma once
5 :
6 : #include <array>
7 : #include <cstddef>
8 :
9 : #include "DataStructures/DataBox/Tag.hpp"
10 : #include "DataStructures/Matrix.hpp"
11 : #include "DataStructures/Tensor/TypeAliases.hpp"
12 : #include "Domain/FaceNormal.hpp"
13 : #include "Evolution/Systems/RelativisticEuler/Valencia/Tags.hpp"
14 : #include "PointwiseFunctions/GeneralRelativity/TagsDeclarations.hpp"
15 : #include "PointwiseFunctions/Hydro/TagsDeclarations.hpp"
16 : #include "Utilities/Gsl.hpp"
17 : #include "Utilities/TMPL.hpp"
18 :
19 : /// \cond
20 : class DataVector;
21 : namespace Tags {
22 : template <typename Tag>
23 : struct Normalized;
24 : } // namespace Tags
25 : /// \endcond
26 :
27 : // IWYU pragma: no_forward_declare Tensor
28 :
29 : namespace RelativisticEuler {
30 : namespace Valencia {
31 :
32 : /// @{
33 : /*!
34 : * \brief Compute the characteristic speeds for the Valencia formulation
35 : * of the relativistic Euler system.
36 : *
37 : * The principal symbol of the system is diagonalized so that the elements of
38 : * the diagonal matrix are the \f$(\text{Dim} + 2)\f$ characteristic speeds
39 : *
40 : * \f{align*}
41 : * \lambda_1 &= \alpha \Lambda^- - \beta_n,\\
42 : * \lambda_{i + 1} &= \alpha v_n - \beta_n,\quad i = 1,...,\text{Dim}\\
43 : * \lambda_{\text{Dim} + 2} &= \alpha \Lambda^+ - \beta_n,
44 : * \f}
45 : *
46 : * where \f$\alpha\f$ is the lapse, \f$\beta_n = n_i \beta^i\f$ and
47 : * \f$v_n = n_i v^i\f$ are the projections of the shift \f$\beta^i\f$ and the
48 : * spatial velocity \f$v^i\f$ onto the normal one-form \f$n_i\f$, respectively,
49 : * and
50 : *
51 : * \f{align*}
52 : * \Lambda^{\pm} &= \dfrac{1}{1 - v^2 c_s^2}\left[ v_n (1- c_s^2) \pm
53 : * c_s\sqrt{\left(1 - v^2\right)\left[1 - v^2 c_s^2 - v_n^2(1 - c_s^2)\right]}
54 : * \right],
55 : * \f}
56 : *
57 : * where \f$v^2 = \gamma_{ij}v^iv^j\f$ is the magnitude squared of the spatial
58 : * velocity, and \f$c_s\f$ is the sound speed.
59 : */
60 : template <size_t Dim>
61 1 : void characteristic_speeds(
62 : gsl::not_null<std::array<DataVector, Dim + 2>*> char_speeds,
63 : const Scalar<DataVector>& lapse, const tnsr::I<DataVector, Dim>& shift,
64 : const tnsr::I<DataVector, Dim>& spatial_velocity,
65 : const Scalar<DataVector>& spatial_velocity_squared,
66 : const Scalar<DataVector>& sound_speed_squared,
67 : const tnsr::i<DataVector, Dim>& normal);
68 :
69 : template <size_t Dim>
70 1 : std::array<DataVector, Dim + 2> characteristic_speeds(
71 : const Scalar<DataVector>& lapse, const tnsr::I<DataVector, Dim>& shift,
72 : const tnsr::I<DataVector, Dim>& spatial_velocity,
73 : const Scalar<DataVector>& spatial_velocity_squared,
74 : const Scalar<DataVector>& sound_speed_squared,
75 : const tnsr::i<DataVector, Dim>& normal);
76 : /// @}
77 :
78 : /*!
79 : * \brief Right eigenvectors of the Valencia formulation
80 : *
81 : * The principal symbol of the Valencia formulation, \f$A\f$, is diagonalizable,
82 : * allowing the introduction of characteristic variables \f$U_\text{char}\f$.
83 : * In terms of the conservative variables \f$U\f$, \f$U = RU_\text{char}\f$ and
84 : * \f$U_\text{char} = LU\f$, where \f$R\f$ is the matrix whose columns are the
85 : * right eigenvectors of \f$A\f$, and \f$L = R^{-1}\f$ contains the left
86 : * eigenvectors as rows. Here we take the ordering
87 : * \f$U = [\tilde D, \tilde \tau, \tilde S_i]^T\f$. Explicitly,
88 : * \f$R = [R_-\, R_1\, R_2\, R_3\, R_+]\f$, where [Teukolsky, to be published]
89 : *
90 : * \f{align*}
91 : * R_\pm = \left[\begin{array}{c}
92 : * 1 \\
93 : * (hW - 1) \pm \dfrac{hW c_s v_n}{d} \\
94 : * hW\left(v_i \pm \dfrac{c_s}{d}n_i\right)
95 : * \end{array}\right],\qquad
96 : * R_{1,2} = \left[\begin{array}{c}
97 : * Wv_{(1,2)} \\
98 : * W\left(2hW - 1\right)v_{(1,2)} \\
99 : * h\left(t_{(1,2)i} + 2W^2v_{(1,2)}v_i \right)
100 : * \end{array}\right],\qquad
101 : * R_3 = \left[\begin{array}{c}
102 : * K \\
103 : * K(hW - 1) - hWc_s^2 \\
104 : * hW(K - c_s^2)v_i
105 : * \end{array}\right].
106 : * \f}
107 : *
108 : * (One disregards \f$R_2\f$ if working in 2-d, and also \f$R_1\f$ if
109 : * working in 1-d, so that the number of eigenvectors is
110 : * always \f$\text{Dim} + 2\f$.) In the above expressions, \f$n_i\f$ is the unit
111 : * spatial normal along which the characteristic decomposition is carried
112 : * out, and \f$t_{(1,2)}^i\f$ are unit spatial vectors forming an orthonormal
113 : * basis with the normal. In addition, \f$h\f$ is the
114 : * specific enthalpy, \f$W\f$ is the Lorentz factor, \f$v^i\f$ is the spatial
115 : * velocity, \f$v_n = n_i v^i\f$, \f$v_{(1,2)} = v_i t^i_{(1,2)}\f$,
116 : * \f$c_s\f$ is the sound speed,
117 : * \f$d = W\sqrt{1 - v^2c_s^2 - v_n^2(1 - c_s^2)}\f$,
118 : * \f$K = (1/\rho)(\partial p/\partial\epsilon)_\rho\f$, where \f$\rho\f$ is the
119 : * rest mass density, \f$p\f$ is the pressure, and \f$\epsilon\f$ is the
120 : * specific internal energy. Finally, and for numerical purposes, the quantity
121 : * \f$(hW - 1)\f$ is computed independently using the expression
122 : *
123 : * \f{align*}
124 : * hW - 1 = W\left(\epsilon + \frac{p}{\rho} + \frac{Wv^2}{W + 1}\right),
125 : * \f}
126 : *
127 : * which has a well-behaved Newtonian limit.
128 : *
129 : * Following the notation in Valencia::characteristic_speeds, \f$R_\pm\f$
130 : * are the eigenvectors of the nondegenerate eigenvalues
131 : * \f$(\alpha\Lambda^\pm - \beta_n)\f$, while \f$ R_{1,2,3}\f$ are the
132 : * eigenvectors of the Dim-fold degenerate eigenvalue
133 : * \f$(\alpha v_n - \beta_n)\f$.
134 : */
135 : template <size_t Dim>
136 1 : Matrix right_eigenvectors(const Scalar<double>& rest_mass_density,
137 : const tnsr::I<double, Dim>& spatial_velocity,
138 : const Scalar<double>& specific_internal_energy,
139 : const Scalar<double>& pressure,
140 : const Scalar<double>& specific_enthalpy,
141 : const Scalar<double>& kappa_over_density,
142 : const Scalar<double>& sound_speed_squared,
143 : const Scalar<double>& lorentz_factor,
144 : const tnsr::ii<double, Dim>& spatial_metric,
145 : const tnsr::II<double, Dim>& inv_spatial_metric,
146 : const Scalar<double>& det_spatial_metric,
147 : const tnsr::i<double, Dim>& unit_normal);
148 :
149 : /*!
150 : * \brief Left eigenvectors of the Valencia formulation
151 : *
152 : * The principal symbol of the Valencia formulation, \f$A\f$, is diagonalizable,
153 : * allowing the introduction of characteristic variables \f$U_\text{char}\f$.
154 : * In terms of the conservative variables \f$U\f$, \f$U = RU_\text{char}\f$ and
155 : * \f$U_\text{char} = LU\f$, where \f$R\f$ is the matrix whose columns are the
156 : * right eigenvectors of \f$A\f$, and \f$L = R^{-1}\f$ contains the left
157 : * eigenvectors as rows. Here we take the ordering
158 : * \f$U = [\tilde D, \tilde \tau, \tilde S_i]^T\f$. Explicitly,
159 : * \f$L = [L_-\, L_1\, L_2\, L_3\, L_+]^T\f$, where [Teukolsky, to be published]
160 : *
161 : * \f{align*}
162 : * L_\pm &= \dfrac{1}{2h W c_s^2(1 - v_n^2)}\left[\begin{array}{ccc}
163 : * W(1-v_n^2)\left[c_s^2(h + W) - K(h - W)\right] - c_\pm, &
164 : * b_\pm, &
165 : * -a v^i + c_s\left(c_s v_n \pm d\right)n^i
166 : * \end{array}\right],\\
167 : * L_{1,2} &= \dfrac{1}{h\left(1 - v_n^2\right)}\left[\begin{array}{ccc}
168 : * -v_{(1,2)}, & -v_{(1,2)}, & v_{(1,2)}v_n n^i + (1 - v_n^2)t_{(1,2)}^i
169 : * \end{array}\right],\\
170 : * L_3 &= \dfrac{1}{hc_s^2}\left[\begin{array}{ccc}
171 : * h - W, & -W, & Wv^i
172 : * \end{array}\right],
173 : * \f}
174 : *
175 : * with the definitions
176 : *
177 : * \f{align*}
178 : * d &= W\sqrt{1 - v^2c_s^2 - v_n^2(1 - c_s^2)}, \\
179 : * a &= W^2\left(1- v_n^2\right)\left(K + c_s^2\right), \\
180 : * c_\pm &= c_s\left(c_s \pm v_nd\right), \\
181 : * b_\pm &= a - c_\pm.
182 : * \f}
183 : *
184 : * The rest of the variables are the same as those in
185 : * Valencia::right_eigenvectors. For numerical purposes, here
186 : * the quantity \f$(h - W)\f$ is computed independently using the expression
187 : *
188 : * \f{align*}
189 : * h - W =\epsilon + \frac{p}{\rho} - \frac{W^2v^2}{W + 1},
190 : * \f}
191 : *
192 : * which has a well-behaved Newtonian limit.
193 : *
194 : * Following the notation in Valencia::characteristic_speeds, \f$L_\pm\f$
195 : * are the eigenvectors of the nondegenerate eigenvalues
196 : * \f$(\alpha\Lambda^\pm - \beta_n)\f$, while \f$ L_{1,2,3}\f$ are the
197 : * eigenvectors of the Dim-fold degenerate eigenvalue
198 : * \f$(\alpha v_n - \beta_n)\f$.
199 : */
200 : template <size_t Dim>
201 1 : Matrix left_eigenvectors(const Scalar<double>& rest_mass_density,
202 : const tnsr::I<double, Dim>& spatial_velocity,
203 : const Scalar<double>& specific_internal_energy,
204 : const Scalar<double>& pressure,
205 : const Scalar<double>& specific_enthalpy,
206 : const Scalar<double>& kappa_over_density,
207 : const Scalar<double>& sound_speed_squared,
208 : const Scalar<double>& lorentz_factor,
209 : const tnsr::ii<double, Dim>& spatial_metric,
210 : const tnsr::II<double, Dim>& inv_spatial_metric,
211 : const Scalar<double>& det_spatial_metric,
212 : const tnsr::i<double, Dim>& unit_normal);
213 :
214 1 : namespace Tags {
215 : template <size_t Dim>
216 0 : struct CharacteristicSpeedsCompute : Tags::CharacteristicSpeeds<Dim>,
217 : db::ComputeTag {
218 0 : using base = Tags::CharacteristicSpeeds<Dim>;
219 0 : using argument_tags =
220 : tmpl::list<gr::Tags::Lapse<DataVector>, gr::Tags::Shift<DataVector, Dim>,
221 : gr::Tags::SpatialMetric<DataVector, Dim>,
222 : hydro::Tags::SpatialVelocity<DataVector, Dim>,
223 : hydro::Tags::SoundSpeedSquared<DataVector>,
224 : ::Tags::Normalized<domain::Tags::UnnormalizedFaceNormal<Dim>>>;
225 :
226 0 : using volume_tags = tmpl::list<>;
227 :
228 0 : using return_type = std::array<DataVector, Dim + 2>;
229 :
230 0 : static constexpr void function(
231 : const gsl::not_null<return_type*> result, const Scalar<DataVector>& lapse,
232 : const tnsr::I<DataVector, Dim>& shift,
233 : const tnsr::ii<DataVector, Dim>& spatial_metric,
234 : const tnsr::I<DataVector, Dim>& spatial_velocity,
235 : const Scalar<DataVector>& sound_speed_squared,
236 : const tnsr::i<DataVector, Dim>& unit_normal) {
237 : characteristic_speeds(
238 : result, lapse, shift, spatial_velocity,
239 : dot_product(spatial_velocity, spatial_velocity, spatial_metric),
240 : sound_speed_squared, unit_normal);
241 : }
242 : };
243 :
244 : } // namespace Tags
245 :
246 0 : struct ComputeLargestCharacteristicSpeed {
247 0 : using argument_tags = tmpl::list<>;
248 0 : static double apply() { return 1.0; }
249 : };
250 :
251 : } // namespace Valencia
252 : } // namespace RelativisticEuler
|
