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Tempus::StepperIMEX_RK_Partition< Scalar > Class Template Reference

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 >:
Tempus::StepperImplicit< Scalar > Tempus::StepperRKBase< Scalar > Tempus::Stepper< Scalar > Tempus::Stepper< Scalar > Teuchos::Describable Teuchos::VerboseObject< Stepper< Scalar > > Teuchos::Describable Teuchos::VerboseObject< Stepper< Scalar > > Teuchos::LabeledObject Teuchos::VerboseObjectBase Teuchos::LabeledObject Teuchos::VerboseObjectBase

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 VerboseObjectsetVerbLevel (const EVerbosityLevel verbLevel) const
 
virtual const VerboseObjectsetOverridingVerbLevel (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 VerboseObjectBasesetOStream (const RCP< FancyOStream > &oStream) const
 
virtual const VerboseObjectBasesetOverridingOStream (const RCP< FancyOStream > &oStream) const
 
virtual VerboseObjectBasesetLinePrefix (const std::string &linePrefix)
 
virtual RCP< FancyOStreamgetOStream () const
 
virtual RCP< FancyOStreamgetOverridingOStream () 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< FancyOStreamgetDefaultOStream ()
 
- 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, $W$. 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

  1. 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.
  2. 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
template<class Scalar >
virtual bool Tempus::StepperIMEX_RK_Partition< Scalar >::isImplicit ( ) const
inlinevirtual
template<class Scalar >
virtual bool Tempus::StepperIMEX_RK_Partition< Scalar >::isExplicitImplicit ( ) const
inlinevirtual
template<class Scalar >
virtual bool Tempus::StepperIMEX_RK_Partition< Scalar >::isOneStepMethod ( ) const
inlinevirtual
template<class Scalar >
virtual bool Tempus::StepperIMEX_RK_Partition< Scalar >::isMultiStepMethod ( ) const
inlinevirtual
template<class Scalar >
virtual OrderODE Tempus::StepperIMEX_RK_Partition< Scalar >::getOrderODE ( ) const
inlinevirtual
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

Definition at line 610 of file Tempus_StepperIMEX_RK_Partition_impl.hpp.

template<class Scalar >
void Tempus::StepperIMEX_RK_Partition< Scalar >::setOrder ( Scalar  order)
inline

Definition at line 458 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

Member Data Documentation

template<class Scalar >
Teuchos::RCP<const RKButcherTableau<Scalar> > Tempus::StepperIMEX_RK_Partition< Scalar >::explicitTableau_
protected

Definition at line 461 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 462 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.

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

Definition at line 464 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 466 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 467 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 469 of file Tempus_StepperIMEX_RK_Partition_decl.hpp.


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