#include <source/navier_stokes/parabolic_module.h>

Inheritance diagram for ryujin::NavierStokes::ParabolicModule< dim, Number >:

Collaboration diagram for ryujin::NavierStokes::ParabolicModule< dim, Number >:

Public Member Functions
Constructor and setup
	ParabolicModule (const MPIEnsemble &mpi_ensemble, const OfflineData< dim, Number > &offline_data, const HyperbolicSystem &hyperbolic_system, const ParabolicSystem &parabolic_system, const InitialValues< Description, dim, Number > &initial_values, const std::string &subsection="ParabolicModule")

void	prepare ()

Functions for performing explicit time steps
void	reinit_state_vector (StateVector &) const

void	prepare_state_vector (StateVector &state_vector, Number t) const

template<int stages>
void	backward_euler_step (const StateVector &old_state_vector, const Number old_t, std::array< std::reference_wrapper< const StateVector >, stages > stage_state_vectors, const std::array< Number, stages > stage_weights, StateVector &new_state_vector, Number tau) const

void	crank_nicolson_step (const StateVector &old_state_vector, const Number old_t, StateVector &new_state_vector, Number tau) const

void	set_id_violation_strategy (const IDViolationStrategy &strategy) const

Information and statistics
void	print_solver_statistics (std::ostream &output) const

Typedefs and constexpr constants
using	HyperbolicSystem = Euler::HyperbolicSystem

using	View = Euler::HyperbolicSystemView< dim, Number >

using	ParabolicSystem = NavierStokes::ParabolicSystem

using	StateVector = typename View::StateVector

using	ScalarHostVector = Vectors::ScalarHostVector< Number >

using	BlockHostVector = Vectors::BlockHostVector< Number >

using	ScalarNumber = typename View::ScalarNumber

using	state_type = typename View::state_type

static constexpr auto	problem_dimension = View::problem_dimension

Detailed Description

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

Implicit backward-Euler 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 implicit backward Euler step updating the velocity (see [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+1} + \dt\sum_{j\in\calI(i)} \polB_{ij} \bsfV^{n+1} = m_i \bsfM_i^{n} + \dt m_i \bsfF_i^{n+1}, & \forall i\in \calV\upint \\[0.3em] \bsfV_i^{n+1} = \boldsymbol 0, & \forall i\in \calV\upbnd, \end{cases} \end{align}

We then postprocess and compute an internal energy update with an additional backward Euler step, (cf. [10], Eq. 5.13)

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

The result is then transformed back into conserved quantities and written to the output vector.

Note: The backward Euler scheme is a fundamental building block for higher-order time stepping, including the well-known Crank-Nicolson scheme. The latter can be expressed algebraically as a backward Euler step (from time \(t\) to \(t+\tau\) followed by an extrapolation step \(U^{n+2}=2U^{n+1}-U^{n}\) from time \(t+\tau\) to \(t+2\tau\)). This approach differs from the Crank-Nicolson scheme discussed in [10] where the extrapolation step is performed on the primitive quantities (velocity and internal energy) instead of the conserved quantities.

Definition at line 114 of file parabolic_module.h.

Member Typedef Documentation

◆ HyperbolicSystem

template<int dim, typename Number = double>

using ryujin::NavierStokes::ParabolicModule< dim, Number >::HyperbolicSystem = Euler::HyperbolicSystem

Definition at line 122 of file parabolic_module.h.

◆ View

template<int dim, typename Number = double>

using ryujin::NavierStokes::ParabolicModule< dim, Number >::View = Euler::HyperbolicSystemView<dim, Number>

Definition at line 124 of file parabolic_module.h.

◆ ParabolicSystem

template<int dim, typename Number = double>

using ryujin::NavierStokes::ParabolicModule< dim, Number >::ParabolicSystem = NavierStokes::ParabolicSystem

Definition at line 126 of file parabolic_module.h.

◆ StateVector

template<int dim, typename Number = double>

using ryujin::NavierStokes::ParabolicModule< dim, Number >::StateVector = typename View::StateVector

Definition at line 128 of file parabolic_module.h.

◆ ScalarHostVector

template<int dim, typename Number = double>

using ryujin::NavierStokes::ParabolicModule< dim, Number >::ScalarHostVector = Vectors::ScalarHostVector<Number>

Definition at line 130 of file parabolic_module.h.

◆ BlockHostVector

template<int dim, typename Number = double>

using ryujin::NavierStokes::ParabolicModule< dim, Number >::BlockHostVector = Vectors::BlockHostVector<Number>

Definition at line 132 of file parabolic_module.h.

◆ ScalarNumber

template<int dim, typename Number = double>

using ryujin::NavierStokes::ParabolicModule< dim, Number >::ScalarNumber = typename View::ScalarNumber

Definition at line 134 of file parabolic_module.h.

◆ state_type

template<int dim, typename Number = double>

using ryujin::NavierStokes::ParabolicModule< dim, Number >::state_type = typename View::state_type

Definition at line 138 of file parabolic_module.h.

Constructor & Destructor Documentation

◆ ParabolicModule()

template<int dim, typename Number >

ryujin::NavierStokes::ParabolicModule< dim, Number >::ParabolicModule	(	const MPIEnsemble &	mpi_ensemble,
		const OfflineData< dim, Number > &	offline_data,
		const HyperbolicSystem &	hyperbolic_system,
		const ParabolicSystem &	parabolic_system,
		const InitialValues< Description, dim, Number > &	initial_values,
		const std::string &	subsection = `"ParabolicModule< dim, Number >"`
	)

Constructor.

Definition at line 33 of file parabolic_module.template.h.

Member Function Documentation

◆ prepare()

template<int dim, typename Number >

void ryujin::NavierStokes::ParabolicModule< dim, Number >::prepare ( )

Prepare time stepping. A call to prepare() allocates temporary storage and is necessary before any of the following time-stepping functions can be called.

Definition at line 125 of file parabolic_module.template.h.

References ryujin::cg_q1, ryujin::dirichlet, and ryujin::no_slip.

◆ reinit_state_vector()

template<int dim, typename Number = double>

void ryujin::NavierStokes::ParabolicModule< dim, Number >::reinit_state_vector ( StateVector & ) const

inline

(Re)initialize the parabolic state vector component of the state vector.

Note: This routine does not modify the hyperbolic state vector or the precomputed vector component.

Definition at line 177 of file parabolic_module.h.

◆ prepare_state_vector()

template<int dim, typename Number >

void ryujin::NavierStokes::ParabolicModule< dim, Number >::prepare_state_vector	(	StateVector &	state_vector,
		Number	t
	)		const

Prepare the parabolic part of the state vector prior to a backward Euler or Crank Nicolson step.

Definition at line 224 of file parabolic_module.template.h.

◆ backward_euler_step()

template<int dim, typename Number >

template<int stages>

void ryujin::NavierStokes::ParabolicModule< dim, Number >::backward_euler_step	(	const StateVector &	old_state_vector,
		const Number	old_t,
		std::array< std::reference_wrapper< const StateVector >, stages >	stage_state_vectors,
		const std::array< Number, stages >	stage_weights,
		StateVector &	new_state_vector,
		Number	tau
	)		const

Given a reference to a previous state vector old_U at time old_t and a time-step size tau perform an implicit backward euler step (and store the result in new_U).

The function takes an optional array of states stage_U together with a an array of weights stage_weights to construct a modified high-order right-hand side / flux.

Definition at line 236 of file parabolic_module.template.h.

◆ crank_nicolson_step()

template<int dim, typename Number >

void ryujin::NavierStokes::ParabolicModule< dim, Number >::crank_nicolson_step	(	const StateVector &	old_state_vector,
		const Number	old_t,
		StateVector &	new_state_vector,
		Number	tau
	)		const

Given a reference to a previous state vector old_state_vector at time old_t and a time-step size tau perform an implicit Crank Nicolson step (and store the result in new_state_vector).

Definition at line 256 of file parabolic_module.template.h.

◆ set_id_violation_strategy()

template<int dim, typename Number = double>

void ryujin::NavierStokes::ParabolicModule< dim, Number >::set_id_violation_strategy ( const IDViolationStrategy & strategy ) const

inline

Sets the invariant domain violation strategy.

Definition at line 220 of file parabolic_module.h.

◆ print_solver_statistics()

template<int dim, typename Number >

void ryujin::NavierStokes::ParabolicModule< dim, Number >::print_solver_statistics ( std::ostream & output ) const

Print a status line with solver statistics. This function is used for constructing the status message displayed periodically in the TimeLoop.

Definition at line 289 of file parabolic_module.template.h.

Member Data Documentation

◆ problem_dimension

template<int dim, typename Number = double>

constexpr auto ryujin::NavierStokes::ParabolicModule< dim, Number >::problem_dimension = View::problem_dimension

staticconstexpr

Definition at line 136 of file parabolic_module.h.

The documentation for this class was generated from the following files:

source/navier_stokes/parabolic_module.h
source/navier_stokes/parabolic_module.template.h

Public Member Functions

Typedefs and constexpr constants

Detailed Description

Member Typedef Documentation

◆ HyperbolicSystem

◆ View

◆ ParabolicSystem

◆ StateVector

◆ ScalarHostVector

◆ BlockHostVector

◆ ScalarNumber

◆ state_type

Constructor & Destructor Documentation

◆ ParabolicModule()

Member Function Documentation

◆ prepare()

◆ reinit_state_vector()

◆ prepare_state_vector()

◆ backward_euler_step()

◆ crank_nicolson_step()

◆ set_id_violation_strategy()

◆ print_solver_statistics()

Member Data Documentation

◆ problem_dimension