Partitioned Implicit-Explicit Runge-Kutta (IMEX-RK) time stepper. More...

#include <Tempus_StepperIMEX_RK_Partition_decl.hpp>

Inheritance diagram for Tempus::StepperIMEX_RK_Partition< Scalar >:

Public Member Functions
	StepperIMEX_RK_Partition (std::string stepperType="Partitioned IMEX RK SSP2")
	Default constructor. More...

	StepperIMEX_RK_Partition (const Teuchos::RCP< const Thyra::ModelEvaluator< Scalar > > &appModel, const Teuchos::RCP< Thyra::NonlinearSolverBase< Scalar > > &solver, bool useFSAL, std::string ICConsistency, bool ICConsistencyCheck, bool zeroInitialGuess, const Teuchos::RCP< StepperRKAppAction< Scalar > > &stepperRKAppAction, std::string stepperType, Teuchos::RCP< const RKButcherTableau< Scalar > > explicitTableau, Teuchos::RCP< const RKButcherTableau< Scalar > > implicitTableau, Scalar order)
	Constructor to for all member data. More...

std::vector< Teuchos::RCP < Thyra::VectorBase< Scalar > > > &	getStageF ()

std::vector< Teuchos::RCP < Thyra::VectorBase< Scalar > > > &	getStageGx ()

Teuchos::RCP < Thyra::VectorBase< Scalar > > &	getXTilde ()

virtual Scalar	getAlpha (const Scalar dt) const
	Return alpha = d(xDot)/dx. More...

virtual Scalar	getBeta (const Scalar) const
	Return beta = d(x)/dx. More...

Teuchos::RCP< const Teuchos::ParameterList >	getValidParameters () const

virtual bool	isValidSetup (Teuchos::FancyOStream &out) const

void	evalImplicitModelExplicitly (const Teuchos::RCP< const Thyra::VectorBase< Scalar > > &X, const Teuchos::RCP< const Thyra::VectorBase< Scalar > > &Y, Scalar time, Scalar stepSize, Scalar stageNumber, const Teuchos::RCP< Thyra::VectorBase< Scalar > > &G) const

void	evalExplicitModel (const Teuchos::RCP< const Thyra::VectorBase< Scalar > > &X, Scalar time, Scalar stepSize, Scalar stageNumber, const Teuchos::RCP< Thyra::VectorBase< Scalar > > &F) const

void	setOrder (Scalar order)

Public Member Functions inherited from Tempus::StepperImplicit< Scalar >
Teuchos::RCP < Teuchos::ParameterList >	getValidParametersBasicImplicit () const

void	setStepperImplicitValues (Teuchos::RCP< Teuchos::ParameterList > pl)
	Set StepperImplicit member data from the ParameterList. More...

void	setStepperSolverValues (Teuchos::RCP< Teuchos::ParameterList > pl)
	Set solver from ParameterList. More...

void	setSolverName (std::string i)
	Set the Solver Name. More...

std::string	getSolverName () const
	Get the Solver Name. More...

virtual Teuchos::RCP< const WrapperModelEvaluator< Scalar > >	getWrapperModel ()

virtual void	setDefaultSolver ()

virtual void	setSolver (Teuchos::RCP< Thyra::NonlinearSolverBase< Scalar > > solver) override
	Set solver. More...

virtual Teuchos::RCP < Thyra::NonlinearSolverBase < Scalar > >	getSolver () const override
	Get solver. More...

const Thyra::SolveStatus< Scalar >	solveImplicitODE (const Teuchos::RCP< Thyra::VectorBase< Scalar > > &x, const Teuchos::RCP< Thyra::VectorBase< Scalar > > &xDot, const Scalar time, const Teuchos::RCP< ImplicitODEParameters< Scalar > > &p, const Teuchos::RCP< Thyra::VectorBase< Scalar > > &y=Teuchos::null, const int index=-1)
	Solve implicit ODE, f(x, xDot, t, p) = 0. More...

void	evaluateImplicitODE (Teuchos::RCP< Thyra::VectorBase< Scalar > > &f, const Teuchos::RCP< Thyra::VectorBase< Scalar > > &x, const Teuchos::RCP< Thyra::VectorBase< Scalar > > &xDot, const Scalar time, const Teuchos::RCP< ImplicitODEParameters< Scalar > > &p)
	Evaluate implicit ODE residual, f(x, xDot, t, p). More...

virtual void	setInitialGuess (Teuchos::RCP< const Thyra::VectorBase< Scalar > > initialGuess) override
	Pass initial guess to Newton solver (only relevant for implicit solvers) More...

virtual void	setZeroInitialGuess (bool zIG)

virtual bool	getZeroInitialGuess () const

virtual Scalar	getInitTimeStep (const Teuchos::RCP< SolutionHistory< Scalar > > &) const override

Public Member Functions inherited from Tempus::Stepper< Scalar >
void	setStepperValues (const Teuchos::RCP< Teuchos::ParameterList > pl)
	Set Stepper member data from ParameterList. More...

Teuchos::RCP < Teuchos::ParameterList >	getValidParametersBasic () const
	Add basic parameters to Steppers ParameterList. More...

virtual bool	isInitialized ()
	True if stepper's member data is initialized. More...

virtual void	checkInitialized ()
	Check initialization, and error out on failure. More...

void	setStepperName (std::string s)
	Set the stepper name. More...

std::string	getStepperName () const
	Get the stepper name. More...

std::string	getStepperType () const
	Get the stepper type. The stepper type is used as an identifier for the stepper, and can only be set by the derived Stepper class. More...

virtual void	setUseFSAL (bool a)

void	setUseFSALTrueOnly (bool a)

void	setUseFSALFalseOnly (bool a)

bool	getUseFSAL () const

void	setICConsistency (std::string s)

std::string	getICConsistency () const

void	setICConsistencyCheck (bool c)

bool	getICConsistencyCheck () const

virtual Teuchos::RCP < Thyra::VectorBase< Scalar > >	getStepperX ()
	Get Stepper x. More...

virtual Teuchos::RCP < Thyra::VectorBase< Scalar > >	getStepperXDot ()
	Get Stepper xDot. More...

virtual Teuchos::RCP < Thyra::VectorBase< Scalar > >	getStepperXDotDot ()
	Get Stepper xDotDot. More...

virtual Teuchos::RCP < Thyra::VectorBase< Scalar > >	getStepperXDotDot (Teuchos::RCP< SolutionState< Scalar > > state)
	Get xDotDot from SolutionState or Stepper storage. More...

virtual std::string	description () const

Public Member Functions inherited from Teuchos::Describable
void	describe (std::ostream &out, const EVerbosityLevel verbLevel=verbLevel_default) const

virtual	~Describable ()

	LabeledObject ()

virtual	~LabeledObject ()

virtual void	setObjectLabel (const std::string &objectLabel)

virtual std::string	getObjectLabel () const

DescribableStreamManipulatorState	describe (const Describable &describable, const EVerbosityLevel verbLevel=Describable::verbLevel_default)

std::ostream &	operator<< (std::ostream &os, const DescribableStreamManipulatorState &d)

Public Member Functions inherited from Teuchos::VerboseObject< Stepper< Scalar > >
	VerboseObject (const EVerbosityLevel verbLevel=VERB_DEFAULT, const RCP< FancyOStream > &oStream=Teuchos::null)

virtual const VerboseObject &	setVerbLevel (const EVerbosityLevel verbLevel) const

virtual const VerboseObject &	setOverridingVerbLevel (const EVerbosityLevel verbLevel) const

virtual EVerbosityLevel	getVerbLevel () const

TEUCHOSPARAMETERLIST_LIB_DLL_EXPORT RCP< const ParameterList >	getValidVerboseObjectSublist ()

TEUCHOSPARAMETERLIST_LIB_DLL_EXPORT void	setupVerboseObjectSublist (ParameterList *paramList)

TEUCHOSPARAMETERLIST_LIB_DLL_EXPORT void	readVerboseObjectSublist (ParameterList paramList, RCP< FancyOStream > oStream, EVerbosityLevel *verbLevel)

void	readVerboseObjectSublist (ParameterList paramList, VerboseObject< ObjectType > verboseObject)

Public Member Functions inherited from Teuchos::VerboseObjectBase
virtual	~VerboseObjectBase ()

	VerboseObjectBase (const RCP< FancyOStream > &oStream=Teuchos::null)

virtual const VerboseObjectBase &	setOStream (const RCP< FancyOStream > &oStream) const

virtual const VerboseObjectBase &	setOverridingOStream (const RCP< FancyOStream > &oStream) const

virtual VerboseObjectBase &	setLinePrefix (const std::string &linePrefix)

virtual RCP< FancyOStream >	getOStream () const

virtual RCP< FancyOStream >	getOverridingOStream () const

virtual std::string	getLinePrefix () const

virtual OSTab	getOSTab (const int tabs=1, const std::string &linePrefix="") const

Public Member Functions inherited from Tempus::StepperRKBase< Scalar >
virtual int	getNumberOfStages () const

virtual int	getStageNumber () const

virtual void	setStageNumber (int s)

virtual void	setUseEmbedded (bool a)

virtual bool	getUseEmbedded () const

virtual void	setErrorNorm (const Teuchos::RCP< Stepper_ErrorNorm< Scalar >> &errCalculator=Teuchos::null)

virtual void	setAppAction (Teuchos::RCP< StepperRKAppAction< Scalar >> appAction)

virtual Teuchos::RCP < StepperRKAppAction< Scalar > >	getAppAction () const

virtual void	setStepperRKValues (Teuchos::RCP< Teuchos::ParameterList > pl)
	Set StepperRK member data from the ParameterList. More...

virtual Teuchos::RCP < RKButcherTableau< Scalar > >	createTableau (Teuchos::RCP< Teuchos::ParameterList > pl)

Protected Attributes
Teuchos::RCP< const RKButcherTableau< Scalar > >	explicitTableau_

Teuchos::RCP< const RKButcherTableau< Scalar > >	implicitTableau_

Scalar	order_

std::vector< Teuchos::RCP < Thyra::VectorBase< Scalar > > >	stageF_

std::vector< Teuchos::RCP < Thyra::VectorBase< Scalar > > >	stageGx_

Teuchos::RCP < Thyra::VectorBase< Scalar > >	xTilde_

Protected Attributes inherited from Tempus::StepperImplicit< Scalar >
Teuchos::RCP < WrapperModelEvaluator < Scalar > >	wrapperModel_

Teuchos::RCP < Thyra::NonlinearSolverBase < Scalar > >	solver_

Teuchos::RCP< const Thyra::VectorBase< Scalar > >	initialGuess_

bool	zeroInitialGuess_

std::string	solverName_

Protected Attributes inherited from Tempus::Stepper< Scalar >
bool	useFSAL_ = false
	Use First-Same-As-Last (FSAL) principle. More...

bool	isInitialized_
	True if stepper's member data is initialized. More...

Protected Attributes inherited from Tempus::StepperRKBase< Scalar >
Teuchos::RCP< RKButcherTableau < Scalar > >	tableau_

bool	useEmbedded_

Teuchos::RCP < Thyra::VectorBase< Scalar > >	ee_

Teuchos::RCP < Thyra::VectorBase< Scalar > >	abs_u0

Teuchos::RCP < Thyra::VectorBase< Scalar > >	abs_u

Teuchos::RCP < Thyra::VectorBase< Scalar > >	sc

Teuchos::RCP < Stepper_ErrorNorm< Scalar > >	stepperErrorNormCalculator_

int	stageNumber_

Teuchos::RCP < StepperRKAppAction< Scalar > >	stepperRKAppAction_

Basic stepper methods
virtual Teuchos::RCP< const RKButcherTableau< Scalar > >	getTableau () const
	Returns the explicit tableau! More...

virtual void	setTableaus (std::string stepperType="", Teuchos::RCP< const RKButcherTableau< Scalar > > explicitTableau=Teuchos::null, Teuchos::RCP< const RKButcherTableau< Scalar > > implicitTableau=Teuchos::null)
	Set both the explicit and implicit tableau from ParameterList. More...

virtual void	setTableausPartition (Teuchos::RCP< Teuchos::ParameterList > pl, std::string stepperType)

virtual Teuchos::RCP< const RKButcherTableau< Scalar > >	getExplicitTableau () const
	Return explicit tableau. More...

virtual void	setExplicitTableau (Teuchos::RCP< const RKButcherTableau< Scalar > > explicitTableau)
	Set the explicit tableau from tableau. More...

virtual Teuchos::RCP< const RKButcherTableau< Scalar > >	getImplicitTableau () const
	Return implicit tableau. More...

virtual void	setImplicitTableau (Teuchos::RCP< const RKButcherTableau< Scalar > > implicitTableau)
	Set the implicit tableau from tableau. More...

virtual void	setModel (const Teuchos::RCP< const Thyra::ModelEvaluator< Scalar > > &appModel)
	Set the model. More...

virtual Teuchos::RCP< const Thyra::ModelEvaluator< Scalar > >	getModel () const

virtual void	setModelPair (const Teuchos::RCP< WrapperModelEvaluatorPairPartIMEX_Basic< Scalar > > &modelPair)
	Create WrapperModelPairIMEX from user-supplied ModelEvaluator pair. More...

virtual void	setModelPair (const Teuchos::RCP< const Thyra::ModelEvaluator< Scalar > > &explicitModel, const Teuchos::RCP< const Thyra::ModelEvaluator< Scalar > > &implicitModel)
	Create WrapperModelPairIMEX from explicit/implicit ModelEvaluators. More...

virtual void	initialize ()
	Initialize during construction and after changing input parameters. More...

virtual void	setInitialConditions (const Teuchos::RCP< SolutionHistory< Scalar > > &solutionHistory)
	Set the initial conditions and make them consistent. More...

virtual void	takeStep (const Teuchos::RCP< SolutionHistory< Scalar > > &solutionHistory)
	Take the specified timestep, dt, and return true if successful. More...

virtual Teuchos::RCP < Tempus::StepperState< Scalar > >	getDefaultStepperState ()
	Provide a StepperState to the SolutionState. This Stepper does not have any special state data, so just provide the base class StepperState with the Stepper description. This can be checked to ensure that the input StepperState can be used by this Stepper. More...

virtual Scalar	getOrder () const

virtual Scalar	getOrderMin () const

virtual Scalar	getOrderMax () const

virtual bool	isExplicit () const

virtual bool	isImplicit () const

virtual bool	isExplicitImplicit () const

virtual bool	isOneStepMethod () const

virtual bool	isMultiStepMethod () const

virtual OrderODE	getOrderODE () const

Overridden from Teuchos::Describable
virtual void	describe (Teuchos::FancyOStream &out, const Teuchos::EVerbosityLevel verbLevel) const

Additional Inherited Members
Static Public Member Functions inherited from Teuchos::VerboseObject< Stepper< Scalar > >
static void	setDefaultVerbLevel (const EVerbosityLevel defaultVerbLevel)

static EVerbosityLevel	getDefaultVerbLevel ()

Static Public Member Functions inherited from Teuchos::VerboseObjectBase
static void	setDefaultOStream (const RCP< FancyOStream > &defaultOStream)

static RCP< FancyOStream >	getDefaultOStream ()

Static Public Attributes inherited from Teuchos::Describable
static const EVerbosityLevel	verbLevel_default

Protected Member Functions inherited from Tempus::Stepper< Scalar >
virtual void	setStepperX (Teuchos::RCP< Thyra::VectorBase< Scalar > > x)
	Set x for Stepper storage. More...

virtual void	setStepperXDot (Teuchos::RCP< Thyra::VectorBase< Scalar > > xDot)
	Set xDot for Stepper storage. More...

virtual void	setStepperXDotDot (Teuchos::RCP< Thyra::VectorBase< Scalar > > xDotDot)
	Set x for Stepper storage. More...

void	setStepperType (std::string s)
	Set the stepper type. More...

Protected Member Functions inherited from Teuchos::VerboseObject< Stepper< Scalar > >
void	initializeVerboseObject (const EVerbosityLevel verbLevel=VERB_DEFAULT, const RCP< FancyOStream > &oStream=Teuchos::null)

Protected Member Functions inherited from Teuchos::VerboseObjectBase
void	initializeVerboseObjectBase (const RCP< FancyOStream > &oStream=Teuchos::null)

virtual void	informUpdatedVerbosityState () const

Protected Member Functions inherited from Tempus::StepperRKBase< Scalar >
virtual void	setEmbeddedMemory ()

Detailed Description

template<class Scalar>
class Tempus::StepperIMEX_RK_Partition< Scalar >

Partitioned Implicit-Explicit Runge-Kutta (IMEX-RK) time stepper.

Partitioned IMEX-RK is similar to the IMEX-RK (StepperIMEX_RK), except a portion of the solution only requires explicit integration, and should not be part of the implicit solution to reduce computational costs. Again our implicit ODE can be written as

$\begin{eqnarray*} M(z,t)\, \dot{z} + G(z,t) + F(z,t) & = & 0, \\ \mathcal{G}(\dot{z},z,t) + F(z,t) & = & 0, \end{eqnarray*}$

but now

$z =\left\{\begin{array}{c} y\\ x \end{array}\right\},\; F(z,t)=\left\{\begin{array}{c} F^y(x,y,t)\\ F^x(x,y,t)\end{array}\right\}, \mbox{ and } G(z,t)=\left\{\begin{array}{c} 0\\ G^x(x,y,t) \end{array}\right\}$

where $z$ is the product vector of $y$ and $x$ , $F(z,t)$ is still the "slow" physics (and evolved explicitly), and $G(z,t)$ is still the "fast" physics (and evolved implicitly), but a portion of the solution vector, $y$ , is "explicit-only" and is only evolved by $F^y(x,y,t)$ , while $x$ is the Implicit/Explicit (IMEX) solution vector, and is evolved explicitly by $F^x(x,y,t)$ and is evolved implicitly by $G^x(x,y,t)$ . Note we can expand this to show all the terms as

$\begin{eqnarray*} & & M^y(x,y,t)\: \dot{y} + F^y(x,y,t) = 0, \\ & & M^x(x,y,t)\: \dot{x} + F^x(x,y,t) + G^x(x,y,t) = 0, \\ \end{eqnarray*}$

or

$\left\{ \begin{array}{c} \dot{y} \\ \dot{x} \end{array}\right\} + \left\{ \begin{array}{c} f^y \\ f^x \end{array}\right\} + \left\{ \begin{array}{c} 0 \\ g^x \end{array}\right\} = 0$

where

$\begin{eqnarray*} f^y(x,y,t) & = & M^y(x,y,t)^{-1}\, F^y(x,y,t), \\ f^x(x,y,t) & = & M^x(x,y,t)^{-1}\, F^x(x,y,t), \\ g^x(x,y,t) & = & M^x(x,y,t)^{-1}\, G^x(x,y,t), \end{eqnarray*}$

or

$\dot{z} + g(x,y,t) + f(x,y,t) = 0,$

where $f(x,y,t) = M(x,y,t)^{-1}\, F(x,y,t)$ , and $g(x,y,t) = M(x,y,t)^{-1}\, G(x,y,t)$ . Using Butcher tableaus for the explicit and implicit terms

$\begin{array}{c|c} \hat{c} & \hat{a} \\ \hline & \hat{b}^T \end{array} \;\;\;\; \mbox{ and } \;\;\;\; \begin{array}{c|c} c & a \\ \hline & b^T \end{array},$

respectively, the basic scheme for this partitioned, $s$ -stage, IMEX-RK method is

$\begin{array}{rcll} Z_i & = & Z_{n-1} - \Delta t \sum_{j=1}^{i-1} \hat{a}_{ij}\; f(Z_j,\hat{t}_j) - \Delta t \sum_{j=1}^i a_{ij}\; g(Z_j, t_j) & \mbox{for } i=1\ldots s, \\ z_n & = & z_{n-1} - \Delta t \sum_{i=1}^s \left[ \hat{b}_i\; f(Z_i,\hat{t}_i) + b_i\; g(Z_i, t_i) \right] & \end{array}$

or expanded

$\begin{array}{rcll} Y_i & = & y_{n-1} - \Delta t \sum_{j=1}^{i-1} \hat{a}_{ij}\; f^y(Z_j,\hat{t}_j) & \mbox{for } i=1\ldots s,\\ X_i & = & x_{n-1} - \Delta t \sum_{j=1}^{i-1} \hat{a}_{ij}\; f^x(Z_j,\hat{t}_j) - \Delta t \sum_{j=1}^i a_{ij}\; g^x(Z_j, t_j) & \mbox{for } i=1\ldots s, \\ y_n & = & y_{n-1} - \Delta t \sum_{i=1}^s \hat{b}_{i}\; f^y(Z_i,\hat{t}_i) & \\ x_n & = & x_{n-1} - \Delta t \sum_{i=1}^s \left[ \hat{b}_i\; f^x(Z_i,\hat{t}_i) + b_i\; g^x(Z_i, t_i) \right] & \end{array}$

where $\hat{t}_i = t_{n-1}+\hat{c}_i\Delta t$ and $t_i = t_{n-1}+c_i\Delta t$ . Note that the "slow" explicit physics, $f^y(Z_j,\hat{t}_j)$ and $f^x(Z_j,\hat{t}_j)$ , is evaluated at the explicit stage time, $\hat{t}_j$ , and the "fast" implicit physics, $g^x(Z_j,t_j)$ , is evaluated at the implicit stage time, $t_j$ . We can write the stage solution, $Z_i$ , as

$Z_i = \tilde{Z} - a_{ii} \Delta t\, g(Z_i,t_i)$

where

$\tilde{Z} = z_{n-1} - \Delta t \sum_{j=1}^{i-1} \left[\hat{a}_{ij}\, f(Z_j,\hat{t}_j) + a_{ij}\, g(Z_j, t_j)\right]$

or in expanded form as

$\left\{ \begin{array}{c} Y_i \\ X_i \end{array}\right\} = \left\{ \begin{array}{c} \tilde{Y} \\ \tilde{X}_i \end{array}\right\} - a_{ii} \Delta t \left\{ \begin{array}{c} 0 \\ g^x(Z_i,t_i) \end{array}\right\}$

where

$\begin{eqnarray*} \tilde{Y} & = & y_{n-1} - \Delta t \sum_{j=1}^{i-1} \left[\hat{a}_{ij}\, f^y(Z_j,\hat{t}_j)\right] \\ \tilde{X} & = & x_{n-1} - \Delta t \sum_{j=1}^{i-1} \left[\hat{a}_{ij}\, f^x(Z_j,\hat{t}_j) +a_{ij}\, g^x(Z_j,t_j)\right] \\ \end{eqnarray*}$

and note that $Y_i = \tilde{Y}$ .

Noting that we will be solving the implicit ODE for $\dot{X}_i$ , we can write

$\mathcal{G}^x(\tilde{\dot{X}},X_i,Y_i,t_i) = \tilde{\dot{X}} + g^x(X_i,Y_i,t_i) = 0$

where we have defined a pseudo time derivative, $\tilde{\dot{X}}$ ,

$\tilde{\dot{X}} \equiv \frac{X_i - \tilde{X}}{a_{ii} \Delta t} \quad \quad \left[ = -g^x(X_i,Y_i,t_i)\right]$

that can be used with the implicit solve but is not the stage time derivative for the IMEX equations, $\dot{X}_i$ . (Note that $\tilde{\dot{X}}$ can be interpreted as the rate of change of the solution due to the implicit "fast" physics.) Note that we are solving for $X_i$ , and $Y_i$ are included as parameters possibly needed in the IMEX equations.

To obtain the stage time derivative, $\dot{Z}_i$ , for the entire system, we can evaluate the governing equation at the implicit stage time, $t_i$ ,

$\dot{Z}_i(Z_i,t_i) + f(Z_i,t_i) + g(Z_i,t_i) = 0$

The above time derivative, $\dot{Z}_i$ , is likely not the same as the real time derivative, $\dot{x}(x(t_i), y(t_i), t_i)$ , unless $\hat{c}_i = c_i \rightarrow \hat{t}_i = t_i$ (Reasoning: $x(t_i) \neq X_i$ and $y(t_i) \neq Y_i$ unless $\hat{t}_i = t_i$ ). Also note that the explicit term, $f(Z_i,t_i)$ , is evaluated at the implicit stage time, $t_i$ . Solving for $\dot{Z}_i$ , we find

$\dot{Z}(Z_i,t_i) = - g(Z_i,t_i) - f(Z_i,t_i)$

Iteration Matrix, . Recalling that the definition of the iteration matrix, $W$ , is

$W = \alpha \frac{\partial \mathcal{F}_n}{\partial \dot{x}_n} + \beta \frac{\partial \mathcal{F}_n}{\partial x_n},$

where $\alpha \equiv \frac{\partial \dot{x}_n(x_n) }{\partial x_n},$ and $\beta \equiv \frac{\partial x_n}{\partial x_n} = 1$ . For the IMEX equations, we are solving

$\mathcal{G}^x(\tilde{\dot{X}},X_i,Y_i,t_i) = \tilde{\dot{X}} + g^x(X_i,Y_i,t_i) = 0$

where $\mathcal{F}_n \rightarrow \mathcal{G}^x$ , $x_n \rightarrow X_{i}$ , and $\dot{x}_n(x_n) \rightarrow \tilde{\dot{X}}(X_{i})$ . The time derivative for the implicit solves is

$\tilde{\dot{X}} \equiv \frac{X_i - \tilde{X}}{a_{ii} \Delta t}$

and we can determine that $\alpha = \frac{1}{a_{ii} \Delta t}$ and $\beta = 1$ , and therefore write

$W = \frac{1}{a_{ii} \Delta t} \frac{\partial \mathcal{G}^x}{\partial \tilde{\dot{X}}} + \frac{\partial \mathcal{G}^x}{\partial X_i}.$

Explicit Stage in the Implicit Tableau. For general DIRK methods, we need to also handle the case when $a_{ii}=0$ . The IMEX stage values can be simply evaluated similiar to the "explicit-only" stage values, e.g.,

$X_i = \tilde{X} = x_{n-1} - \Delta t \sum_{j=1}^{i-1} \left[\hat{a}_{ij}\, f^x(Z_j,\hat{t}_j) +a_{ij}\, g^x(Z_j,t_j)\right]$

and the time derivative of the stage solution is

$\dot{X}_i = - g^x(X_i,Y_i,t_i) - f^x(X_i,Y_i,t_i)$

but again note that the explicit term, $f^x(X_i,Y_i,t_i)$ , is evaluated at the implicit stage time, $t_i$ .

Algorithm The single-timestep algorithm for the partitioned IMEX-RK is

$\begin{center} \parbox{5in}{ \rule{5in}{0.4pt} \\ {\bf Algorithm} Partitioned IMEX-RK \\ \rule{5in}{0.4pt} \vspace{-15pt} \begin{enumerate} \setlength{\itemsep}{0pt} \setlength{\parskip}{0pt} \setlength{\parsep}{0pt} \item $Z \leftarrow z_{n-1}$ \hfill {\it * Recall $Z_i = \{Y_i,X_i\}^T$} \item {\it appAction.execute(solutionHistory, stepper, BEGIN\_STEP)} \item {\bf for ($i = 0 \ldots s-1$)} \item \quad $Y_i = y_{n-1} -\Delta t \sum_{j=1}^{i-1} \hat{a}_{ij}\;f^y_j$ \item \quad $\tilde{X} \leftarrow x_{n-1} - \Delta t\,\sum_{j=1}^{i-1} \left[ \hat{a}_{ij}\, f^x_j + a_{ij}\, g^x_j \right]$ \item \quad {\it appAction.execute(solutionHistory, stepper, BEGIN\_STAGE)} \item \quad \hfill {\bf Implicit Tableau} \item \quad {\bf if ($a_{ii} = 0$) then} \item \qquad $X_i \leftarrow \tilde{X}$ \item \qquad {\bf if ($a_{k,i} = 0 \;\forall k = (i+1,\ldots, s-1)$, $b(i) = 0$, $b^\ast(i) = 0$) then} \item \qquad \quad $g^x_i \leftarrow 0$ \hfill {\it * Not needed for later calculations.} \item \qquad {\bf else} \item \qquad \quad $g^x_i \leftarrow g^x(X_i,Y_i,t_i)$ \item \qquad {\bf endif} \item \quad {\bf else} \item \qquad {\it appAction.execute(solutionHistory, stepper, BEFORE\_SOLVE)} \item \qquad {\bf if (``Zero initial guess.'') then} \item \qquad \quad $X \leftarrow 0$ \hfill {\it * Else use previous stage value as initial guess.} \item \qquad {\bf endif} \item \qquad {\bf Solve $\mathcal{G}^x\left(\tilde{\dot{X}} = \frac{X-\tilde{X}}{a_{ii} \Delta t},X,Y,t_i\right) = 0$ for $X$ where $Y$ is known.} \item \qquad {\it appAction.execute(solutionHistory, stepper, AFTER\_SOLVE)} \item \qquad $\tilde{\dot{X}} \leftarrow \frac{X - \tilde{X}}{a_{ii} \Delta t}$ \item \qquad $g_i \leftarrow - \tilde{\dot{X}}$ \item \quad {\bf endif} \item \quad \hfill {\bf Explicit Tableau} \item \quad {\it appAction.execute(solutionHistory, stepper, BEFORE\_EXPLICIT\_EVAL)} \item \quad $f_i \leftarrow M(X,\hat{t}_i)^{-1}\, F(X,\hat{t}_i)$ \item \quad $\dot{Z} \leftarrow - g(Z_i,t_i) - f(Z_i,t_i)$ [Optionally] \item \quad {\it appAction.execute(solutionHistory, stepper, END\_STAGE)} \item {\bf end for} \item $z_n = z_{n-1} - \Delta t\,\sum_{i=1}^{s}\hat{b}_i\, f_i$ \item $x_n \mathrel{+{=}} - \Delta t\,\sum_{i=1}^{s} b_i\, g^x_i$ \item {\it appAction.execute(solutionHistory, stepper, END\_STEP)} \end{enumerate} \vspace{-10pt} \rule{5in}{0.4pt} } \end{center}$

The following table contains the pre-coded IMEX-RK tableaus.

Partitioned IMEX-RK Tableaus
Name	Order	Implicit Tableau	Explicit Tableau
Partitioned IMEX RK 1st order	1st	$\begin{array}{c\|cc} 0 & 0 & 0 \\ 1 & 0 & 1 \\ \hline & 0 & 1 \end{array}$	$\begin{array}{c\|cc} 0 & 0 & 0 \\ 1 & 1 & 0 \\ \hline & 1 & 0 \end{array}$
Partitioned IMEX RK SSP2 $\gamma = 1-1/\sqrt{2}$	2nd	$\begin{array}{c\|cc} \gamma & \gamma & 0 \\ 1-\gamma & 1-2\gamma & \gamma \\ \hline & 1/2 & 1/2 \end{array}$	$\begin{array}{c\|cc} 0 & 0 & 0 \\ 1 & 1 & 0 \\ \hline & 1/2 & 1/2 \end{array}$
Partitioned IMEX RK ARS 233 $\gamma = (3+\sqrt{3})/6$	3rd	$\begin{array}{c\|ccc} 0 & 0 & 0 & 0 \\ \gamma & 0 & \gamma & 0 \\ 1-\gamma & 0 & 1-2\gamma & \gamma \\ \hline & 0 & 1/2 & 1/2 \end{array}$	$\begin{array}{c\|ccc} 0 & 0 & 0 & 0 \\ \gamma & \gamma & 0 & 0 \\ 1-\gamma & \gamma-1 & 2-2\gamma & 0 \\ \hline & 0 & 1/2 & 1/2 \end{array}$

The First-Same-As-Last (FSAL) principle is not valid for IMEX RK Partition. The default is to set useFSAL=false, and useFSAL=true will result in a warning.

References

Shadid, Cyr, Pawlowski, Widley, Scovazzi, Zeng, Phillips, Conde, Chuadhry, Hensinger, Fischer, Robinson, Rider, Niederhaus, Sanchez, "Towards an IMEX Monolithic ALE Method with Integrated UQ for Multiphysics Shock-hydro", SAND2016-11353, 2016, pp. 21-28.
Cyr, "IMEX Lagrangian Methods", SAND2015-3745C.

Definition at line 311 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

Constructor & Destructor Documentation

template<class Scalar >

Tempus::StepperIMEX_RK_Partition< Scalar >::StepperIMEX_RK_Partition ( std::string stepperType = "Partitioned IMEX RK SSP2" )

Default constructor.

Requires subsequent setModel(), setSolver() and initialize() calls before calling takeStep().

Definition at line 21 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<class Scalar >

Tempus::StepperIMEX_RK_Partition< Scalar >::StepperIMEX_RK_Partition	(	const Teuchos::RCP< const Thyra::ModelEvaluator< Scalar > > &	appModel,
		const Teuchos::RCP< Thyra::NonlinearSolverBase< Scalar > > &	solver,
		bool	useFSAL,
		std::string	ICConsistency,
		bool	ICConsistencyCheck,
		bool	zeroInitialGuess,
		const Teuchos::RCP< StepperRKAppAction< Scalar > > &	stepperRKAppAction,
		std::string	stepperType,
		Teuchos::RCP< const RKButcherTableau< Scalar > >	explicitTableau,
		Teuchos::RCP< const RKButcherTableau< Scalar > >	implicitTableau,
		Scalar	order
	)

Constructor to for all member data.

Definition at line 54 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

Member Function Documentation

template<class Scalar >

virtual Teuchos::RCP<const RKButcherTableau<Scalar> > Tempus::StepperIMEX_RK_Partition< Scalar >::getTableau ( ) const

inlinevirtual

Returns the explicit tableau!

Reimplemented from Tempus::StepperRKBase< Scalar >.

Definition at line 337 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

void Tempus::StepperIMEX_RK_Partition< Scalar >::setTableaus	(	std::string	stepperType = `""`,
		Teuchos::RCP< const RKButcherTableau< Scalar > >	explicitTableau = `Teuchos::null`,
		Teuchos::RCP< const RKButcherTableau< Scalar > >	implicitTableau = `Teuchos::null`
	)

virtual

Set both the explicit and implicit tableau from ParameterList.

Definition at line 106 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<class Scalar >

void Tempus::StepperIMEX_RK_Partition< Scalar >::setTableausPartition	(	Teuchos::RCP< Teuchos::ParameterList >	pl,
		std::string	stepperType
	)

virtual

Definition at line 334 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<class Scalar >

virtual Teuchos::RCP<const RKButcherTableau<Scalar> > Tempus::StepperIMEX_RK_Partition< Scalar >::getExplicitTableau ( ) const

inlinevirtual

Return explicit tableau.

Definition at line 354 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

void Tempus::StepperIMEX_RK_Partition< Scalar >::setExplicitTableau ( Teuchos::RCP< const RKButcherTableau< Scalar > > explicitTableau )

virtual

Set the explicit tableau from tableau.

Definition at line 399 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<class Scalar >

virtual Teuchos::RCP<const RKButcherTableau<Scalar> > Tempus::StepperIMEX_RK_Partition< Scalar >::getImplicitTableau ( ) const

inlinevirtual

Return implicit tableau.

Definition at line 365 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

void Tempus::StepperIMEX_RK_Partition< Scalar >::setImplicitTableau ( Teuchos::RCP< const RKButcherTableau< Scalar > > implicitTableau )

virtual

Set the implicit tableau from tableau.

Definition at line 412 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<class Scalar >

void Tempus::StepperIMEX_RK_Partition< Scalar >::setModel ( const Teuchos::RCP< const Thyra::ModelEvaluator< Scalar > > & appModel )

virtual

Set the model.

Reimplemented from Tempus::StepperImplicit< Scalar >.

Definition at line 425 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<class Scalar >

virtual Teuchos::RCP<const Thyra::ModelEvaluator<Scalar> > Tempus::StepperIMEX_RK_Partition< Scalar >::getModel ( ) const

inlinevirtual

Reimplemented from Tempus::StepperImplicit< Scalar >.

Definition at line 378 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

void Tempus::StepperIMEX_RK_Partition< Scalar >::setModelPair ( const Teuchos::RCP< WrapperModelEvaluatorPairPartIMEX_Basic< Scalar > > & mePairIMEX )

virtual

Create WrapperModelPairIMEX from user-supplied ModelEvaluator pair.

The user-supplied ME pair can contain any user-specific IMEX interactions between explicit and implicit MEs.

Definition at line 454 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<class Scalar >

void Tempus::StepperIMEX_RK_Partition< Scalar >::setModelPair	(	const Teuchos::RCP< const Thyra::ModelEvaluator< Scalar > > &	explicitModel,
		const Teuchos::RCP< const Thyra::ModelEvaluator< Scalar > > &	implicitModel
	)

virtual

Create WrapperModelPairIMEX from explicit/implicit ModelEvaluators.

Use the supplied explicit/implicit MEs to create a WrapperModelPairIMEX with basic IMEX interactions between explicit and implicit MEs.

Definition at line 478 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<class Scalar >

void Tempus::StepperIMEX_RK_Partition< Scalar >::initialize ( )

virtual

Initialize during construction and after changing input parameters.

Reimplemented from Tempus::Stepper< Scalar >.

Definition at line 492 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<class Scalar >

void Tempus::StepperIMEX_RK_Partition< Scalar >::setInitialConditions ( const Teuchos::RCP< SolutionHistory< Scalar > > & solutionHistory )

virtual

Set the initial conditions and make them consistent.

Reimplemented from Tempus::StepperImplicit< Scalar >.

Definition at line 524 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<class Scalar >

void Tempus::StepperIMEX_RK_Partition< Scalar >::takeStep ( const Teuchos::RCP< SolutionHistory< Scalar > > & solutionHistory )

virtual

Take the specified timestep, dt, and return true if successful.

Implements Tempus::Stepper< Scalar >.

Definition at line 645 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<class Scalar >

Teuchos::RCP< Tempus::StepperState< Scalar > > Tempus::StepperIMEX_RK_Partition< Scalar >::getDefaultStepperState ( )

virtual

Provide a StepperState to the SolutionState. This Stepper does not have any special state data, so just provide the base class StepperState with the Stepper description. This can be checked to ensure that the input StepperState can be used by this Stepper.

Implements Tempus::Stepper< Scalar >.

Definition at line 827 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<class Scalar >

virtual Scalar Tempus::StepperIMEX_RK_Partition< Scalar >::getOrder ( ) const

inlinevirtual

Reimplemented from Tempus::StepperRKBase< Scalar >.

Definition at line 403 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

virtual Scalar Tempus::StepperIMEX_RK_Partition< Scalar >::getOrderMin ( ) const

inlinevirtual

Reimplemented from Tempus::StepperRKBase< Scalar >.

Definition at line 404 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

virtual Scalar Tempus::StepperIMEX_RK_Partition< Scalar >::getOrderMax ( ) const

inlinevirtual

Reimplemented from Tempus::StepperRKBase< Scalar >.

Definition at line 405 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

virtual bool Tempus::StepperIMEX_RK_Partition< Scalar >::isExplicit ( ) const

inlinevirtual

Implements Tempus::Stepper< Scalar >.

Definition at line 407 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

virtual bool Tempus::StepperIMEX_RK_Partition< Scalar >::isImplicit ( ) const

inlinevirtual

Implements Tempus::Stepper< Scalar >.

Definition at line 408 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

virtual bool Tempus::StepperIMEX_RK_Partition< Scalar >::isExplicitImplicit ( ) const

inlinevirtual

Implements Tempus::Stepper< Scalar >.

Definition at line 409 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

virtual bool Tempus::StepperIMEX_RK_Partition< Scalar >::isOneStepMethod ( ) const

inlinevirtual

Implements Tempus::Stepper< Scalar >.

Definition at line 413 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

virtual bool Tempus::StepperIMEX_RK_Partition< Scalar >::isMultiStepMethod ( ) const

inlinevirtual

Implements Tempus::Stepper< Scalar >.

Definition at line 414 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

virtual OrderODE Tempus::StepperIMEX_RK_Partition< Scalar >::getOrderODE ( ) const

inlinevirtual

Implements Tempus::Stepper< Scalar >.

Definition at line 415 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

std::vector<Teuchos::RCP<Thyra::VectorBase<Scalar> > >& Tempus::StepperIMEX_RK_Partition< Scalar >::getStageF ( )

inline

Definition at line 418 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

std::vector<Teuchos::RCP<Thyra::VectorBase<Scalar> > >& Tempus::StepperIMEX_RK_Partition< Scalar >::getStageGx ( )

inline

Definition at line 422 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

Teuchos::RCP<Thyra::VectorBase<Scalar> >& Tempus::StepperIMEX_RK_Partition< Scalar >::getXTilde ( )

inline

Definition at line 426 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

virtual Scalar Tempus::StepperIMEX_RK_Partition< Scalar >::getAlpha ( const Scalar dt ) const

inlinevirtual

Return alpha = d(xDot)/dx.

Implements Tempus::StepperImplicit< Scalar >.

Definition at line 429 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

virtual Scalar Tempus::StepperIMEX_RK_Partition< Scalar >::getBeta ( const Scalar ) const

inlinevirtual

Return beta = d(x)/dx.

Implements Tempus::StepperImplicit< Scalar >.

Definition at line 435 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

Teuchos::RCP< const Teuchos::ParameterList > Tempus::StepperIMEX_RK_Partition< Scalar >::getValidParameters ( ) const

virtual

Reimplemented from Tempus::StepperImplicit< Scalar >.

Definition at line 918 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<class Scalar >

void Tempus::StepperIMEX_RK_Partition< Scalar >::describe	(	Teuchos::FancyOStream &	out,
		const Teuchos::EVerbosityLevel	verbLevel
	)		const

virtual

Reimplemented from Tempus::StepperImplicit< Scalar >.

Definition at line 835 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<class Scalar >

bool Tempus::StepperIMEX_RK_Partition< Scalar >::isValidSetup ( Teuchos::FancyOStream & out ) const

virtual

Reimplemented from Tempus::StepperImplicit< Scalar >.

Definition at line 865 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<typename Scalar >

void Tempus::StepperIMEX_RK_Partition< Scalar >::evalImplicitModelExplicitly	(	const Teuchos::RCP< const Thyra::VectorBase< Scalar > > &	X,
		const Teuchos::RCP< const Thyra::VectorBase< Scalar > > &	Y,
		Scalar	time,
		Scalar	stepSize,
		Scalar	stageNumber,
		const Teuchos::RCP< Thyra::VectorBase< Scalar > > &	G
	)		const

Definition at line 576 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<typename Scalar >

void Tempus::StepperIMEX_RK_Partition< Scalar >::evalExplicitModel	(	const Teuchos::RCP< const Thyra::VectorBase< Scalar > > &	X,
		Scalar	time,
		Scalar	stepSize,
		Scalar	stageNumber,
		const Teuchos::RCP< Thyra::VectorBase< Scalar > > &	F
	)		const