#include <source/navier_stokes/parabolic_solver.h>

Inheritance diagram for ryujin::NavierStokes::ParabolicSolver< Description, dim, Number >:

Collaboration diagram for ryujin::NavierStokes::ParabolicSolver< Description, dim, Number >:

Public Types
using	HyperbolicSystem = typename Description::HyperbolicSystem

using	ParabolicSystem = typename Description::ParabolicSystem

using	HyperbolicSystemView = typename HyperbolicSystem::template View< dim, Number >

using	scalar_type = typename OfflineData< dim, Number >::scalar_type

using	vector_type = typename HyperbolicSystemView::vector_type

using	block_vector_type = dealii::LinearAlgebra::distributed::BlockVector< Number >

Public Member Functions
	ParabolicSolver (const MPI_Comm &mpi_communicator, std::map< std::string, dealii::Timer > &computing_timer, const HyperbolicSystem &hyperbolic_system, const ParabolicSystem &parabolic_system, const OfflineData< dim, Number > &offline_data, const InitialValues< Description, dim, Number > &initial_values, const std::string &subsection="ParabolicSolver")

void	prepare ()

Functions for performing implicit time steps
void	crank_nicolson_step (const vector_type &old_U, const Number old_t, vector_type &new_U, Number tau, const IDViolationStrategy id_violation_strategy, const bool reinitialize_gmg) const

void	print_solver_statistics (std::ostream &output) const

Detailed Description

template<typename Description, int dim, typename Number = double>
class ryujin::NavierStokes::ParabolicSolver< Description, dim, Number >

Minimum entropy guaranteeing second-order time stepping for the parabolic limiting equation [10], Eq. 3.3:

\begin{align} \newcommand{\bbm}{{\boldsymbol m}} \newcommand{\bef}{{\boldsymbol f}} \newcommand{\bk}{{\boldsymbol k}} \newcommand{\bu}{{\boldsymbol u}} \newcommand{\bv}{{\boldsymbol v}} \newcommand{\bn}{{\boldsymbol n}} \newcommand{\pols}{{\mathbb s}} \newcommand{\Hflux}{\bk} &\partial_t \rho = 0, \\ &\partial_t \bbm - \nabla\cdot(\pols(\bv)) = \bef, \\ &\partial_t E + \nabla\cdot(\Hflux(\bu)- \pols(\bv) \bv) = \bef\cdot\bv, \\ &\bv_{|\partial D}=\boldsymbol 0, \qquad \Hflux(\bu)\cdot\bn_{|\partial D}=0 . \end{align}

Internally, the module first performs an explicit second order Crank-Nicolson step updating the velocity [10], Eq. 5.5:

\begin{align} \begin{cases} \newcommand\bsfV{{\textbf V}} \newcommand{\polB}{{\mathbb B}} \newcommand{\calI}{{\mathcal I}} \newcommand\bsfF{{\textbf F}} \newcommand\bsfM{{\textbf M}} \newcommand{\upint}{^\circ} \newcommand{\upbnd}{^\partial} \newcommand{\dt}{{\tau}} \newcommand{\calV}{{\mathcal V}} \varrho^{n}_i m_i \bsfV^{n+\frac{1}{2}} + \tfrac12 \dt\sum_{j\in\calI(i)} \polB_{ij} \bsfV^{n+\frac{1}{2}} = m_i \bsfM_i^{n} + \tfrac12 \dt m_i \bsfF_i^{n+\frac12}, & \forall i\in \calV\upint \\[0.3em] \bsfV_i^{n+\frac{1}{2}} = \boldsymbol 0, & \forall i\in \calV\upbnd, \end{cases} \end{align}

We then postprocess and compute an internal energy update with a second Crank-Nicolson step, [10], Eq. 5.13:

\begin{align} \newcommand\sfe{{\mathsf e}} \newcommand{\upHnph}{^{\text{H},n+\frac{1}{2}}} \newcommand{\calI}{{\mathcal I}} \newcommand\sfK{{\mathsf K}} \newcommand{\calV}{{\mathcal V}} m_i \varrho_i^{n}(\sfe_i{\upHnph} - \sfe_i^{n})+\tfrac12\dt \sum_{j\in\calI(i)} \beta_{ij}\sfe_i{\upHnph} = \tfrac12 \dt m_i\sfK_i^{n+\frac{1}{2}}, \qquad \forall i\in \calV. \end{align}

Todo:: Provide more details about boundary conditions.

Definition at line 101 of file parabolic_solver.h.