Kaskade::SemiImplicitEulerStep< PE > Class Template Reference

Linearly implicit Euler method. More...

#include <semieuler.hh>

Detailed Description

template<class PE>
class Kaskade::SemiImplicitEulerStep< PE >

Linearly implicit Euler method.

A class that defines the weak formulation of the stationary elliptic problem that results from the linearly implicit Euler method for

\[ B \dot u = L(u) , \]

which is, doing the linearization at \( u_0 \),

\[ (B - \tau L'(u_0)) \delta u_k = \tau L(u_k), \quad u_{k+1} = u_k + \delta u_k. \]

The left hand side \( B - \tau L'(u_0) \) is assembled as matrix, and the right hand side \( \tau L(u_k) \) is assembled as right hand side by the assembler, if fed with this weak problem formulation:

assembler.assemble(semiLinearization(SemiImplicitEulerStep,u,uJ,du));

\( B \) may depend on \( u \), made explicit by setting

B2<row,col>::constant = false 

in the parabolic equation class. In this case, \( B(u) \) has to be an invertable Nemyckii operator (a diagonal operator, mapping \( u(x) \) to \( B(x,u(x)) u(x) \). Then the linearly implicit Euler scheme is (see below for derivation)

\[ (B(u_0) - \tau L'(u_0) + \tau_k B^{-1}(u_0)B'(u_0)\mathrm{diag}L(u_0)) \delta u_k = \tau_k L(u)+\frac{\tau_k}{\tau_{k-1}}(B(u_0)-B(u_k))\delta u_{k-1}. \]

The values \( u_0 \) (linearization point for the Jacobian), \( u_k \) (evaluation point for the right hand side), and \( \tau_k/\tau_{k-1}\, \delta u_{k-1} \) (previous increment) have to be specified when constructing the DomainCache. This is most conveniently done via SemiLinearizationAt.

Template Parameters

PE	the weak formulation of the parabolic equation, adhering to the ParabolicEquation interface.

Derivation: Lubich and Roche (Rosenbrock Methods for Differential-algebraic Systems with Solution-dependent Singular Matrix Multiplying the Derivative, Computing 43:325-342, 1990) suggest to reformulate the parabolic system with non-constant \( B \) above as

\[ \begin{aligned} \dot u &= z \\ 0 &= L(u)-B(u)z \end{aligned} \]

and applying, e.g., Rosenbrock methods. Here, simply a linearly implicit Euler. Linearizing the right hand side of the extended system at \( u_0, z_0 \) yields

\[ \Bigg( \begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix} - \tau_k \begin{bmatrix} 0 & I \\ L'(u_0)-B'(u_0)z_0 & -B(u_0) \end{bmatrix} \Bigg) \begin{bmatrix} \delta u_k \\ \delta z_k \end{bmatrix} = \tau_k \begin{bmatrix} z_k \\ L(u_k)-B(u_k)z_k \end{bmatrix}. \]

The first row immediately gives \( \delta u_k = \tau_k (z_k + \delta z_k) \), and the second one

\[ - \tau_k (L'(u_0)-B'(u_0)z_0) \delta u_k + \tau (B(u_0)\delta z_k + B(u_k)z_k) = \tau_k f(u_k). \]

The second term on the left hand side is just

\[ \tau_k (B(u_0)(z_k+\delta z_k) + (B(u_k)-B(u_0))z_k) = B(u_0)\delta u_k + (B(u_k)-B(u_0)) \tau_k z_k, \]

hence we obtain

\[ \left(B(u_0) - \tau_k(L'(u_0)-B'(u_0)z_0)\right)\delta u_k = \tau_k L(u_k)+(B(u_0)-B(u_k)) \frac{\tau_k}{\tau_{k-1}}\delta u_{k-1}. \]

Choosing a consistent initial value \( z_0 = B(u_0)^{-1} L(u_0) \) finally results in the form given above.

Definition at line 197 of file semieuler.hh.

Inheritance diagram for Kaskade::SemiImplicitEulerStep< PE >:

Classes
class	BoundaryCache
struct	D1
struct	D2
class	DomainCache

Public Types
typedef PE	ParabolicEquation
typedef ParabolicEquation::Scalar	Scalar
typedef ParabolicEquation::AnsatzVars	AnsatzVars
typedef ParabolicEquation::TestVars	TestVars
typedef ParabolicEquation::OriginVars	OriginVars
typedef AnsatzVars::Grid	Grid
using	GridView = typename AnsatzVars::GridView
typedef PE	EvolutionEquation
typedef EvolutionEquation::AnsatzVars::template CoefficientVectorRepresentation< 0, EvolutionEquation::TestVars::noOfVariables >::type	CoefficientVectors

Public Member Functions
	SemiImplicitEulerStep (ParabolicEquation *eq_, Scalar dt)
	Constructor. More...
template<class Cell >
int	integrationOrder (Cell const &cell, int shapeFunctionOrder, bool boundary) const
Self &	setTau (Scalar dt)
	Sets the time step size. More...
Scalar	getTau () const
Access to the underlying parabolic equation.
ParabolicEquation &	parabolicEquation ()
ParabolicEquation const &	parabolicEquation () const

Static Public Attributes
static ProblemType const	type = WeakFormulation

Member Typedef Documentation

AnsatzVars

template<class PE >

typedef ParabolicEquation::AnsatzVars Kaskade::SemiImplicitEulerStep< PE >::AnsatzVars

Definition at line 208 of file semieuler.hh.

CoefficientVectors

template<class PE >

typedef EvolutionEquation::AnsatzVars::template CoefficientVectorRepresentation<0,EvolutionEquation::TestVars::noOfVariables>::type Kaskade::SemiImplicitEulerStep< PE >::CoefficientVectors

Definition at line 218 of file semieuler.hh.

EvolutionEquation

template<class PE >

typedef PE Kaskade::SemiImplicitEulerStep< PE >::EvolutionEquation

Definition at line 216 of file semieuler.hh.

Grid

template<class PE >

typedef AnsatzVars::Grid Kaskade::SemiImplicitEulerStep< PE >::Grid

Definition at line 211 of file semieuler.hh.

GridView

template<class PE >

using Kaskade::SemiImplicitEulerStep< PE >::GridView = typename AnsatzVars::GridView

Definition at line 212 of file semieuler.hh.

OriginVars

template<class PE >

typedef ParabolicEquation::OriginVars Kaskade::SemiImplicitEulerStep< PE >::OriginVars

Definition at line 210 of file semieuler.hh.

ParabolicEquation

template<class PE >

typedef PE Kaskade::SemiImplicitEulerStep< PE >::ParabolicEquation

Definition at line 205 of file semieuler.hh.

Scalar

template<class PE >

typedef ParabolicEquation::Scalar Kaskade::SemiImplicitEulerStep< PE >::Scalar

Definition at line 207 of file semieuler.hh.

TestVars

template<class PE >

typedef ParabolicEquation::TestVars Kaskade::SemiImplicitEulerStep< PE >::TestVars

Definition at line 209 of file semieuler.hh.

Constructor & Destructor Documentation

SemiImplicitEulerStep()

template<class PE >

Kaskade::SemiImplicitEulerStep< PE >::SemiImplicitEulerStep ( ParabolicEquation *  eq_, Scalar  dt  )

inline

Constructor.

Parameters

eq	a pointer to a parabolic equation object. This needs to exist during the life time of the SemiImplicitEulerStep object.
dt	the time step. dt>=0 has to hold.

Definition at line 412 of file semieuler.hh.

Member Function Documentation

getTau()

template<class PE >

Scalar Kaskade::SemiImplicitEulerStep< PE >::getTau ( ) const

inline

Definition at line 435 of file semieuler.hh.

Referenced by Kaskade::SemiImplicitEulerStep< PE >::DomainCache::d1(), Kaskade::SemiImplicitEulerStep< PE >::BoundaryCache::d1(), Kaskade::SemiImplicitEulerStep< PE >::DomainCache::d2(), and Kaskade::SemiImplicitEulerStep< PE >::BoundaryCache::d2().

integrationOrder()

template<class PE >

template<class Cell >

int Kaskade::SemiImplicitEulerStep< PE >::integrationOrder ( Cell const &  cell, int  shapeFunctionOrder, bool  boundary  ) const

inline

Definition at line 418 of file semieuler.hh.

parabolicEquation() [1/2]

template<class PE >

ParabolicEquation & Kaskade::SemiImplicitEulerStep< PE >::parabolicEquation ( )

inline

Definition at line 442 of file semieuler.hh.

Referenced by Kaskade::Limex< Eq >::advanceTime(), Kaskade::Limex< Eq >::estimateError(), and Kaskade::Limex< Eq >::step().

parabolicEquation() [2/2]

template<class PE >

ParabolicEquation const & Kaskade::SemiImplicitEulerStep< PE >::parabolicEquation ( ) const

inline

Definition at line 443 of file semieuler.hh.

setTau()

template<class PE >

Self & Kaskade::SemiImplicitEulerStep< PE >::setTau ( Scalar  dt )

inline

Sets the time step size.

Parameters

dt	The new time step size. dt>=0 has to hold.

Definition at line 428 of file semieuler.hh.

Referenced by Kaskade::Limex< Eq >::step().

Member Data Documentation

type

template<class PE >

ProblemType const Kaskade::SemiImplicitEulerStep< PE >::type = WeakFormulation

static

Definition at line 214 of file semieuler.hh.

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

• semieuler.hh