#include <include/ryujin/navier_stokes/parabolic_module.h>

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

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

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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< HyperbolicDescription, dim, Number > &initial_values, const std::string &subsection="ParabolicModule")

void	prepare () override

Functions for performing explicit time steps
void	reinit_state_vector (StateVector< Number > &) const override

void	prepare_state_vector (StateVector< Number > &state_vector, Number t) const override

Number	backward_euler_step (const StateVector< Number > &old_state_vector, StageVectors stage_state_vectors, StageWeights stage_weights, StateVector< Number > &new_state_vector, Number t, Number tau, Number tau_max=std::numeric_limits< Number >::max()) const override

Number	crank_nicolson_step (const StateVector< Number > &old_state_vector, StateVector< Number > &new_state_vector, Number t, Number tau, Number tau_max=std::numeric_limits< Number >::max()) const override

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

Functions for performing implicit time steps

Typedefs and constexpr constants
using	StageVectors = typename Interface::ParabolicModule< dim, Number >::StageVectors

using	StageWeights = typename Interface::ParabolicModule< dim, Number >::StageWeights

using	HyperbolicDescription = Euler::HyperbolicDescription

using	HyperbolicSystem = Euler::HyperbolicSystem

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

using	ParabolicSystem = NavierStokes::ParabolicSystem

using	BlockHostVector = dealii::LinearAlgebra::distributed::BlockVector< 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 [13], 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 [13], 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. [13], 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 [13] where the extrapolation step is performed on the primitive quantities (velocity and internal energy) instead of the conserved quantities.

Definition at line 116 of file parabolic_module.h.

Member Typedef Documentation

◆ StageVectors

template<int dim, typename Number = double>

using ryujin::NavierStokes::ParabolicModule< dim, Number >::StageVectors = typename Interface::ParabolicModule<dim, Number>::StageVectors

Definition at line 124 of file parabolic_module.h.

◆ StageWeights

template<int dim, typename Number = double>

using ryujin::NavierStokes::ParabolicModule< dim, Number >::StageWeights = typename Interface::ParabolicModule<dim, Number>::StageWeights

Definition at line 127 of file parabolic_module.h.

◆ HyperbolicDescription

template<int dim, typename Number = double>

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

Definition at line 130 of file parabolic_module.h.

◆ HyperbolicSystem

template<int dim, typename Number = double>

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

Definition at line 132 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 134 of file parabolic_module.h.

◆ ParabolicSystem

template<int dim, typename Number = double>

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

Definition at line 136 of file parabolic_module.h.

◆ BlockHostVector

template<int dim, typename Number = double>

using ryujin::NavierStokes::ParabolicModule< dim, Number >::BlockHostVector = dealii::LinearAlgebra::distributed::BlockVector<Number>

Definition at line 138 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 141 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 145 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< HyperbolicDescription, dim, Number > &	initial_values,
		const std::string &	subsection = `"ParabolicModule< dim, Number >"`
	)

Constructor.

Definition at line 32 of file parabolic_module.template.h.

Member Function Documentation

◆ prepare()

template<int dim, typename Number >

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

override

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 123 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< Number > & ) const

inlineoverride

(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 185 of file parabolic_module.h.

◆ prepare_state_vector()

template<int dim, typename Number >

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

override

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

Definition at line 222 of file parabolic_module.template.h.

◆ backward_euler_step()

template<int dim, typename Number >

Number ryujin::NavierStokes::ParabolicModule< dim, Number >::backward_euler_step	(	const StateVector< Number > &	old_state_vector,
		StageVectors	stage_state_vectors,
		StageWeights	stage_weights,
		StateVector< Number > &	new_state_vector,
		Number	t,
		Number	tau,
		Number	tau_max = `std::numeric_limits<Number>::max()`
	)		const

overridevirtual

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

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

Implements ryujin::Interface::ParabolicModule< dim, Number >.

Definition at line 233 of file parabolic_module.template.h.

◆ crank_nicolson_step()

template<int dim, typename Number >

Number ryujin::NavierStokes::ParabolicModule< dim, Number >::crank_nicolson_step	(	const StateVector< Number > &	old_state_vector,
		StateVector< Number > &	new_state_vector,
		Number	t,
		Number	tau,
		Number	tau_max = `std::numeric_limits<Number>::max()`
	)		const

overridevirtual

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

Implements ryujin::Interface::ParabolicModule< dim, Number >.

Definition at line 316 of file parabolic_module.template.h.

References ryujin::step.

◆ print_solver_statistics()

template<int dim, typename Number >

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

override

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

Definition at line 356 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 143 of file parabolic_module.h.

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

include/ryujin/navier_stokes/parabolic_module.h
include/ryujin/navier_stokes/parabolic_module.template.h

Public Member Functions

Typedefs and constexpr constants

Detailed Description

Member Typedef Documentation

◆ StageVectors

◆ StageWeights

◆ HyperbolicDescription

◆ HyperbolicSystem

◆ View

◆ ParabolicSystem

◆ BlockHostVector

◆ ScalarNumber

◆ state_type

Constructor & Destructor Documentation

◆ ParabolicModule()

Member Function Documentation

◆ prepare()

◆ reinit_state_vector()

◆ prepare_state_vector()

◆ backward_euler_step()

◆ crank_nicolson_step()

◆ print_solver_statistics()

Member Data Documentation

◆ problem_dimension