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SpECTRE
v2024.04.12
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The Valencia formulation of the relativistic Euler System See Chapter 7 of Relativistic Hydrodynamics by Luciano Rezzolla and Olindo Zanotti or http://iopscience.iop.org/article/10.1086/303604. More...
Namespaces | |
| namespace | BoundaryConditions |
| Boundary conditions for the relativistic Euler system. | |
| namespace | BoundaryCorrections |
| Boundary corrections/numerical fluxes. | |
| namespace | Tags |
| Tags for the Valencia formulation of the relativistic Euler system. | |
Classes | |
| struct | ComputeFluxes |
| The fluxes of the conservative variables. More... | |
| struct | ComputeLargestCharacteristicSpeed |
| struct | ComputeSources |
| Compute the source terms for the flux-conservative Valencia formulation of the relativistic Euler system. More... | |
| struct | ConservativeFromPrimitive |
| Compute the conservative variables from primitive variables. More... | |
| class | FixConservatives |
| Fix conservative variables using the method proposed in F. Foucart's PhD thesis (Cornell) More... | |
| struct | PrimitiveFromConservative |
| Compute the primitive variables from the conservative variables. More... | |
| struct | System |
| struct | TimeDerivativeTerms |
| Compute the time derivative of the conserved variables for the Valencia formulation of the relativistic Euler equations. More... | |
Functions | |
| template<size_t Dim> | |
| Matrix | right_eigenvectors (const Scalar< double > &rest_mass_density, const tnsr::I< double, Dim > &spatial_velocity, const Scalar< double > &specific_internal_energy, const Scalar< double > &pressure, const Scalar< double > &specific_enthalpy, const Scalar< double > &kappa_over_density, const Scalar< double > &sound_speed_squared, const Scalar< double > &lorentz_factor, const tnsr::ii< double, Dim > &spatial_metric, const tnsr::II< double, Dim > &inv_spatial_metric, const Scalar< double > &det_spatial_metric, const tnsr::i< double, Dim > &unit_normal) |
| Right eigenvectors of the Valencia formulation. More... | |
| template<size_t Dim> | |
| Matrix | left_eigenvectors (const Scalar< double > &rest_mass_density, const tnsr::I< double, Dim > &spatial_velocity, const Scalar< double > &specific_internal_energy, const Scalar< double > &pressure, const Scalar< double > &specific_enthalpy, const Scalar< double > &kappa_over_density, const Scalar< double > &sound_speed_squared, const Scalar< double > &lorentz_factor, const tnsr::ii< double, Dim > &spatial_metric, const tnsr::II< double, Dim > &inv_spatial_metric, const Scalar< double > &det_spatial_metric, const tnsr::i< double, Dim > &unit_normal) |
| Left eigenvectors of the Valencia formulation. More... | |
| template<size_t Dim> | |
| bool | operator!= (const FixConservatives< Dim > &lhs, const FixConservatives< Dim > &rhs) |
| template<size_t Dim> | |
| void | characteristic_speeds (gsl::not_null< std::array< DataVector, Dim+2 > * > char_speeds, const Scalar< DataVector > &lapse, const tnsr::I< DataVector, Dim > &shift, const tnsr::I< DataVector, Dim > &spatial_velocity, const Scalar< DataVector > &spatial_velocity_squared, const Scalar< DataVector > &sound_speed_squared, const tnsr::i< DataVector, Dim > &normal) |
| Compute the characteristic speeds for the Valencia formulation of the relativistic Euler system. More... | |
| template<size_t Dim> | |
| std::array< DataVector, Dim+2 > | characteristic_speeds (const Scalar< DataVector > &lapse, const tnsr::I< DataVector, Dim > &shift, const tnsr::I< DataVector, Dim > &spatial_velocity, const Scalar< DataVector > &spatial_velocity_squared, const Scalar< DataVector > &sound_speed_squared, const tnsr::i< DataVector, Dim > &normal) |
| Compute the characteristic speeds for the Valencia formulation of the relativistic Euler system. More... | |
The Valencia formulation of the relativistic Euler System See Chapter 7 of Relativistic Hydrodynamics by Luciano Rezzolla and Olindo Zanotti or http://iopscience.iop.org/article/10.1086/303604.
| std::array< DataVector, Dim+2 > RelativisticEuler::Valencia::characteristic_speeds | ( | const Scalar< DataVector > & | lapse, |
| const tnsr::I< DataVector, Dim > & | shift, | ||
| const tnsr::I< DataVector, Dim > & | spatial_velocity, | ||
| const Scalar< DataVector > & | spatial_velocity_squared, | ||
| const Scalar< DataVector > & | sound_speed_squared, | ||
| const tnsr::i< DataVector, Dim > & | normal | ||
| ) |
Compute the characteristic speeds for the Valencia formulation of the relativistic Euler system.
The principal symbol of the system is diagonalized so that the elements of the diagonal matrix are the \((\text{Dim} + 2)\) characteristic speeds
\begin{align*} \lambda_1 &= \alpha \Lambda^- - \beta_n,\\ \lambda_{i + 1} &= \alpha v_n - \beta_n,\quad i = 1,...,\text{Dim}\\ \lambda_{\text{Dim} + 2} &= \alpha \Lambda^+ - \beta_n, \end{align*}
where \(\alpha\) is the lapse, \(\beta_n = n_i \beta^i\) and \(v_n = n_i v^i\) are the projections of the shift \(\beta^i\) and the spatial velocity \(v^i\) onto the normal one-form \(n_i\), respectively, and
\begin{align*} \Lambda^{\pm} &= \dfrac{1}{1 - v^2 c_s^2}\left[ v_n (1- c_s^2) \pm c_s\sqrt{\left(1 - v^2\right)\left[1 - v^2 c_s^2 - v_n^2(1 - c_s^2)\right]} \right], \end{align*}
where \(v^2 = \gamma_{ij}v^iv^j\) is the magnitude squared of the spatial velocity, and \(c_s\) is the sound speed.
| void RelativisticEuler::Valencia::characteristic_speeds | ( | gsl::not_null< std::array< DataVector, Dim+2 > * > | char_speeds, |
| const Scalar< DataVector > & | lapse, | ||
| const tnsr::I< DataVector, Dim > & | shift, | ||
| const tnsr::I< DataVector, Dim > & | spatial_velocity, | ||
| const Scalar< DataVector > & | spatial_velocity_squared, | ||
| const Scalar< DataVector > & | sound_speed_squared, | ||
| const tnsr::i< DataVector, Dim > & | normal | ||
| ) |
Compute the characteristic speeds for the Valencia formulation of the relativistic Euler system.
The principal symbol of the system is diagonalized so that the elements of the diagonal matrix are the \((\text{Dim} + 2)\) characteristic speeds
\begin{align*} \lambda_1 &= \alpha \Lambda^- - \beta_n,\\ \lambda_{i + 1} &= \alpha v_n - \beta_n,\quad i = 1,...,\text{Dim}\\ \lambda_{\text{Dim} + 2} &= \alpha \Lambda^+ - \beta_n, \end{align*}
where \(\alpha\) is the lapse, \(\beta_n = n_i \beta^i\) and \(v_n = n_i v^i\) are the projections of the shift \(\beta^i\) and the spatial velocity \(v^i\) onto the normal one-form \(n_i\), respectively, and
\begin{align*} \Lambda^{\pm} &= \dfrac{1}{1 - v^2 c_s^2}\left[ v_n (1- c_s^2) \pm c_s\sqrt{\left(1 - v^2\right)\left[1 - v^2 c_s^2 - v_n^2(1 - c_s^2)\right]} \right], \end{align*}
where \(v^2 = \gamma_{ij}v^iv^j\) is the magnitude squared of the spatial velocity, and \(c_s\) is the sound speed.
| Matrix RelativisticEuler::Valencia::left_eigenvectors | ( | const Scalar< double > & | rest_mass_density, |
| const tnsr::I< double, Dim > & | spatial_velocity, | ||
| const Scalar< double > & | specific_internal_energy, | ||
| const Scalar< double > & | pressure, | ||
| const Scalar< double > & | specific_enthalpy, | ||
| const Scalar< double > & | kappa_over_density, | ||
| const Scalar< double > & | sound_speed_squared, | ||
| const Scalar< double > & | lorentz_factor, | ||
| const tnsr::ii< double, Dim > & | spatial_metric, | ||
| const tnsr::II< double, Dim > & | inv_spatial_metric, | ||
| const Scalar< double > & | det_spatial_metric, | ||
| const tnsr::i< double, Dim > & | unit_normal | ||
| ) |
Left eigenvectors of the Valencia formulation.
The principal symbol of the Valencia formulation, \(A\), is diagonalizable, allowing the introduction of characteristic variables \(U_\text{char}\). In terms of the conservative variables \(U\), \(U = RU_\text{char}\) and \(U_\text{char} = LU\), where \(R\) is the matrix whose columns are the right eigenvectors of \(A\), and \(L = R^{-1}\) contains the left eigenvectors as rows. Here we take the ordering \(U = [\tilde D, \tilde \tau, \tilde S_i]^T\). Explicitly, \(L = [L_-\, L_1\, L_2\, L_3\, L_+]^T\), where [Teukolsky, to be published]
\begin{align*} L_\pm &= \dfrac{1}{2h W c_s^2(1 - v_n^2)}\left[\begin{array}{ccc} W(1-v_n^2)\left[c_s^2(h + W) - K(h - W)\right] - c_\pm, & b_\pm, & -a v^i + c_s\left(c_s v_n \pm d\right)n^i \end{array}\right],\\ L_{1,2} &= \dfrac{1}{h\left(1 - v_n^2\right)}\left[\begin{array}{ccc} -v_{(1,2)}, & -v_{(1,2)}, & v_{(1,2)}v_n n^i + (1 - v_n^2)t_{(1,2)}^i \end{array}\right],\\ L_3 &= \dfrac{1}{hc_s^2}\left[\begin{array}{ccc} h - W, & -W, & Wv^i \end{array}\right], \end{align*}
with the definitions
\begin{align*} d &= W\sqrt{1 - v^2c_s^2 - v_n^2(1 - c_s^2)}, \\ a &= W^2\left(1- v_n^2\right)\left(K + c_s^2\right), \\ c_\pm &= c_s\left(c_s \pm v_nd\right), \\ b_\pm &= a - c_\pm. \end{align*}
The rest of the variables are the same as those in Valencia::right_eigenvectors. For numerical purposes, here the quantity \((h - W)\) is computed independently using the expression
\begin{align*} h - W =\epsilon + \frac{p}{\rho} - \frac{W^2v^2}{W + 1}, \end{align*}
which has a well-behaved Newtonian limit.
Following the notation in Valencia::characteristic_speeds, \(L_\pm\) are the eigenvectors of the nondegenerate eigenvalues \((\alpha\Lambda^\pm - \beta_n)\), while \( L_{1,2,3}\) are the eigenvectors of the Dim-fold degenerate eigenvalue \((\alpha v_n - \beta_n)\).
| Matrix RelativisticEuler::Valencia::right_eigenvectors | ( | const Scalar< double > & | rest_mass_density, |
| const tnsr::I< double, Dim > & | spatial_velocity, | ||
| const Scalar< double > & | specific_internal_energy, | ||
| const Scalar< double > & | pressure, | ||
| const Scalar< double > & | specific_enthalpy, | ||
| const Scalar< double > & | kappa_over_density, | ||
| const Scalar< double > & | sound_speed_squared, | ||
| const Scalar< double > & | lorentz_factor, | ||
| const tnsr::ii< double, Dim > & | spatial_metric, | ||
| const tnsr::II< double, Dim > & | inv_spatial_metric, | ||
| const Scalar< double > & | det_spatial_metric, | ||
| const tnsr::i< double, Dim > & | unit_normal | ||
| ) |
Right eigenvectors of the Valencia formulation.
The principal symbol of the Valencia formulation, \(A\), is diagonalizable, allowing the introduction of characteristic variables \(U_\text{char}\). In terms of the conservative variables \(U\), \(U = RU_\text{char}\) and \(U_\text{char} = LU\), where \(R\) is the matrix whose columns are the right eigenvectors of \(A\), and \(L = R^{-1}\) contains the left eigenvectors as rows. Here we take the ordering \(U = [\tilde D, \tilde \tau, \tilde S_i]^T\). Explicitly, \(R = [R_-\, R_1\, R_2\, R_3\, R_+]\), where [Teukolsky, to be published]
\begin{align*} R_\pm = \left[\begin{array}{c} 1 \\ (hW - 1) \pm \dfrac{hW c_s v_n}{d} \\ hW\left(v_i \pm \dfrac{c_s}{d}n_i\right) \end{array}\right],\qquad R_{1,2} = \left[\begin{array}{c} Wv_{(1,2)} \\ W\left(2hW - 1\right)v_{(1,2)} \\ h\left(t_{(1,2)i} + 2W^2v_{(1,2)}v_i \right) \end{array}\right],\qquad R_3 = \left[\begin{array}{c} K \\ K(hW - 1) - hWc_s^2 \\ hW(K - c_s^2)v_i \end{array}\right]. \end{align*}
(One disregards \(R_2\) if working in 2-d, and also \(R_1\) if working in 1-d, so that the number of eigenvectors is always \(\text{Dim} + 2\).) In the above expressions, \(n_i\) is the unit spatial normal along which the characteristic decomposition is carried out, and \(t_{(1,2)}^i\) are unit spatial vectors forming an orthonormal basis with the normal. In addition, \(h\) is the specific enthalpy, \(W\) is the Lorentz factor, \(v^i\) is the spatial velocity, \(v_n = n_i v^i\), \(v_{(1,2)} = v_i t^i_{(1,2)}\), \(c_s\) is the sound speed, \(d = W\sqrt{1 - v^2c_s^2 - v_n^2(1 - c_s^2)}\), \(K = (1/\rho)(\partial p/\partial\epsilon)_\rho\), where \(\rho\) is the rest mass density, \(p\) is the pressure, and \(\epsilon\) is the specific internal energy. Finally, and for numerical purposes, the quantity \((hW - 1)\) is computed independently using the expression
\begin{align*} hW - 1 = W\left(\epsilon + \frac{p}{\rho} + \frac{Wv^2}{W + 1}\right), \end{align*}
which has a well-behaved Newtonian limit.
Following the notation in Valencia::characteristic_speeds, \(R_\pm\) are the eigenvectors of the nondegenerate eigenvalues \((\alpha\Lambda^\pm - \beta_n)\), while \( R_{1,2,3}\) are the eigenvectors of the Dim-fold degenerate eigenvalue \((\alpha v_n - \beta_n)\).