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 ()
	Default constructor. More...

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

Teuchos::RCP < Thyra::VectorBase< Scalar > >	getStageX ()
	Return the full stage solution which is Z (the concat of X and Y) for IMEX Partition. More...

Teuchos::RCP < Thyra::VectorBase< Scalar > >	getStageZ ()
	Explicitly return the full stage solution, Z. 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)

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 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)

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

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	setObserver (Teuchos::RCP< StepperObserver< Scalar > > obs=Teuchos::null)
	Set Observer. More...

virtual Teuchos::RCP < StepperObserver< Scalar > >	getObserver () const
	Get Observer. 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

Public Member Functions inherited from Tempus::StepperImplicit< Scalar >
virtual Teuchos::RCP< const WrapperModelEvaluator< Scalar > >	getWrapperModel ()

virtual void	setDefaultSolver ()

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

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

const Thyra::SolveStatus< Scalar >	solveImplicitODE (const Teuchos::RCP< Thyra::VectorBase< Scalar > > &x)
	Solve problem using x in-place. (Needs to be deprecated!) 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)
	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)
	Pass initial guess to Newton solver (only relevant for implicit solvers) More...

virtual void	setZeroInitialGuess (bool zIG)
	Set parameter so that the initial guess is set to zero (=True) or use last timestep (=False). More...

virtual bool	getZeroInitialGuess () const

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

Public Member Functions inherited from Tempus::Stepper< Scalar >
virtual void	setNonConstModel (const Teuchos::RCP< Thyra::ModelEvaluator< Scalar > > &)

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

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

void	setStepperType (std::string s)

std::string	getStepperType () const

void	setUseFSAL (bool a)

bool	getUseFSAL () const

virtual bool	getUseFSALDefault () const

void	setICConsistency (std::string s)

std::string	getICConsistency () const

virtual std::string	getICConsistencyDefault () const

void	setICConsistencyCheck (bool c)

bool	getICConsistencyCheck () const

virtual bool	getICConsistencyCheckDefault () const

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

virtual Teuchos::RCP < Thyra::VectorBase< Scalar > >	getStepperXDot (Teuchos::RCP< SolutionState< Scalar > > state)
	Get xDot from SolutionState or Stepper storage. 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

virtual void	createSubSteppers (std::vector< Teuchos::RCP< const Thyra::ModelEvaluator< Scalar > > >)

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

virtual int	getStageNumber () const

virtual void	setStageNumber (int s)

virtual Teuchos::RCP< const Thyra::VectorBase< Scalar > >	getStageX () const

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

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

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

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

Scalar	order_

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

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

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

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

Teuchos::RCP < StepperRKObserverComposite < Scalar > >	stepperObserver_

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_

Teuchos::RCP< StepperObserver < Scalar > >	stepperObserver_

Protected Attributes inherited from Tempus::Stepper< Scalar >
bool	isInitialized_ = false
	True if stepper's member data is initialized. More...

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

int	stageNumber_
	The current Runge-Kutta stage number, {0,...,s-1}. -1 indicates outside stage loop. More...

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

Teuchos::RCP < StepperRKAppAction< Scalar > >	stepperRKAppAction_

Additional Inherited Members
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...

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, $t_i$ .

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

$\begin{algorithm} \renewcommand{\thealgorithm}{} \caption{IMEX RK with the application-action locations indicated.} \begin{algorithmic}[1] \State {\it appAction.execute(solutionHistory, stepper, BEGIN\_STEP)} \State $Z \leftarrow z_{n-1}$ (Recall $Z_i = \{Y_i,X_i\}^T$) \Comment Set initial guess to last timestep. \For {$i = 0 \ldots s-1$} \State $Y_i = y_{n-1} -\Delta t \sum_{j=1}^{i-1} \hat{a}_{ij}\;f^y_j$ \State $\tilde{X} \leftarrow x_{n-1} - \Delta t\,\sum_{j=1}^{i-1} \left[ \hat{a}_{ij}\, f^x_j + a_{ij}\, g^x_j \right]$ \State {\it appAction.execute(solutionHistory, stepper, BEGIN\_STAGE)} \State \Comment Implicit Tableau \If {$a_{ii} = 0$} \State $X_i \leftarrow \tilde{X}$ \If {$a_{k,i} = 0 \;\forall k = (i+1,\ldots, s-1)$, $b(i) = 0$, $b^\ast(i) = 0$} \State $g^x_i \leftarrow 0$ \Comment{Not needed for later calculations.} \Else \State $g^x_i \leftarrow g^x(X_i,Y_i,t_i)$ \EndIf \Else \State {\it appAction.execute(solutionHistory, stepper, BEFORE\_SOLVE)} \If {``Zero initial guess.''} \State $X \leftarrow 0$ \Comment{Else use previous stage value as initial guess.} \EndIf \State 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 $Y$ are known parameters \State {\it appAction.execute(solutionHistory, stepper, AFTER\_SOLVE)} \State $\tilde{\dot{X}} \leftarrow \frac{X - \tilde{X}}{a_{ii} \Delta t}$ \State $g_i \leftarrow - \tilde{\dot{X}}$ \EndIf \State \Comment Explicit Tableau \State {\it appAction.execute(solutionHistory, stepper, BEFORE\_EXPLICIT\_EVAL)} \State $f_i \leftarrow M(X,\hat{t}_i)^{-1}\, F(X,\hat{t}_i)$ \State $\dot{Z} \leftarrow - g(Z_i,t_i) - f(Z_i,t_i)$ [Optionally] \State {\it appAction.execute(solutionHistory, stepper, END\_STAGE)} \EndFor \State $z_n = z_{n-1} - \Delta t\,\sum_{i=1}^{s}\hat{b}_i\, f_i$ \State $x_n \mathrel{+{=}} - \Delta t\,\sum_{i=1}^{s} b_i\, g^x_i$ \State {\it appAction.execute(solutionHistory, stepper, END\_STEP)} \end{algorithmic} \end{algorithm}$

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

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 317 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

Constructor & Destructor Documentation

template<class Scalar >

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

Default constructor.

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

Definition at line 27 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< StepperObserver< Scalar > > &	obs,
		const Teuchos::RCP< Thyra::NonlinearSolverBase< Scalar > > &	solver,
		bool	useFSAL,
		std::string	ICConsistency,
		bool	ICConsistencyCheck,
		bool	zeroInitialGuess,
		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 48 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
	)

Definition at line 82 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

Member Function Documentation

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 777 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

Definition at line 534 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 499 of file Tempus_StepperIMEX_RK_Partition_impl.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 441 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 447 of file Tempus_StepperIMEX_RK_Partition_decl.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 768 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 369 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

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

inlinevirtual

Return implicit tableau.

Definition at line 377 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

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

inlinevirtual

Reimplemented from Tempus::StepperImplicit< Scalar >.

Definition at line 387 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

virtual Teuchos::RCP<StepperObserver<Scalar> > Tempus::StepperIMEX_RK_Partition< Scalar >::getObserver ( ) const

inlinevirtual

Get Observer.

Reimplemented from Tempus::Stepper< Scalar >.

Definition at line 402 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

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

inlinevirtual

Reimplemented from Tempus::StepperRKBase< Scalar >.

Definition at line 418 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 420 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 419 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 429 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 436 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 437 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

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

inlinevirtual

Return the full stage solution which is Z (the concat of X and Y) for IMEX Partition.

Reimplemented from Tempus::StepperRKBase< Scalar >.

Definition at line 433 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

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

inline

Explicitly return the full stage solution, Z.

Definition at line 435 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

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 360 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

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

virtual

Implements Tempus::Stepper< Scalar >.

Definition at line 869 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<class Scalar >

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

inline

Definition at line 438 of file Tempus_StepperIMEX_RK_Partition_decl.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 418 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<class Scalar >

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

inlinevirtual

Implements Tempus::Stepper< Scalar >.

Definition at line 422 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 424 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 423 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 427 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 426 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

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

virtual

Reimplemented from Tempus::StepperImplicit< Scalar >.

Definition at line 812 of file Tempus_StepperIMEX_RK_Partition_impl.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 292 of file Tempus_StepperIMEX_RK_Partition_impl.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 306 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 451 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

Reimplemented from Tempus::StepperImplicit< Scalar >.

Definition at line 319 of file Tempus_StepperIMEX_RK_Partition_impl.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 347 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 371 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<class Scalar >

void Tempus::StepperIMEX_RK_Partition< Scalar >::setObserver ( Teuchos::RCP< StepperObserver< Scalar > > obs = Teuchos::null )

virtual

Set Observer.

Reimplemented from Tempus::Stepper< Scalar >.

Definition at line 387 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<class Scalar >

void Tempus::StepperIMEX_RK_Partition< Scalar >::setOrder ( Scalar order )

inline

Definition at line 470 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 120 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 570 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

Member Data Documentation

template<class Scalar >

Teuchos::RCP<const RKButcherTableau<Scalar> > Tempus::StepperIMEX_RK_Partition< Scalar >::explicitTableau_

protected

Definition at line 474 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

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

protected

Definition at line 475 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

Scalar Tempus::StepperIMEX_RK_Partition< Scalar >::order_

protected

Definition at line 477 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

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

protected

Definition at line 480 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

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

protected

Definition at line 481 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

Teuchos::RCP<Thyra::VectorBase<Scalar> > Tempus::StepperIMEX_RK_Partition< Scalar >::stageZ_

protected

Definition at line 479 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

Teuchos::RCP<StepperRKObserverComposite<Scalar> > Tempus::StepperIMEX_RK_Partition< Scalar >::stepperObserver_

protected

Definition at line 486 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

template<class Scalar >

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

protected

Definition at line 483 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

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

http://docs.trilinos.org/r13.0/packages/tempus/src/Tempus_StepperIMEX_RK_Partition_decl.hpp
http://docs.trilinos.org/r13.0/packages/tempus/src/Tempus_StepperIMEX_RK_Partition_impl.hpp

Public Member Functions

Protected Attributes

Additional Inherited Members

Detailed Description

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

References

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

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