spectral time grid for defect correction methods with Radau points on \( [a,b]\). More...

#include <sdc.hh>

Detailed Description

spectral time grid for defect correction methods with Radau points on \( [a,b]\).

For the Radau time grid, \( t_0 = a \) and \( t_n = b \) holds. Since \( t_0 \) is not used for quadrature, there are \( n \) actual quadrature nodes in use. Thus, Radau quadrature as defined here is exact for polynomials of degree up to \( 2n-2 \).

Note that collocation with Radau points is L-stable, and therefore SDC methods on Radau grids are best for highly stiff problems (e.g., parabolic equations or Dirichlet b.c. realized as quadratic penalty). Consider using GaussTimeGrid or LobattoTimeGrid in other cases.

Definition at line 425 of file sdc.hh.

Inheritance diagram for Kaskade::RadauTimeGrid:

Public Types
typedef Dune::DynamicVector< double >	RealVector
	The type used for real vectors. More...

typedef DynamicMatrix< double >	RealMatrix
	The type used for real (dense) matrices. More...

Public Member Functions
	RadauTimeGrid (int n, double a, double b)
	constructs a Radau grid with \( n+1 \) points on \( [a,b] \) More...

virtual void	refine (RealMatrix &p)
	perform refinement of the grid, filling the prolongation matrix More...

virtual RealMatrix	interpolate (RealVector const &x) const
	Compute interpolation coefficients. More...

virtual RealVector const &	points () const
	Time points in the time step. More...

virtual RealMatrix const &	integrationMatrix () const
	Integration matrix \( S \). More...

virtual RealMatrix const &	differentiationMatrix () const
	Differentiation matrix \( D \). More...

double	point (int k) const
	Time points in the time step. More...

Protected Attributes
RealVector	pts

RealMatrix	integ

RealMatrix	diff

Member Typedef Documentation

◆ RealMatrix

typedef DynamicMatrix<double> Kaskade::SDCTimeGrid::RealMatrix

inherited

The type used for real (dense) matrices.

Definition at line 78 of file sdc.hh.

◆ RealVector

typedef Dune::DynamicVector<double> Kaskade::SDCTimeGrid::RealVector

inherited

The type used for real vectors.

Definition at line 73 of file sdc.hh.

Constructor & Destructor Documentation

◆ RadauTimeGrid()

Kaskade::RadauTimeGrid::RadauTimeGrid	(	int	n,
		double	a,
		double	b
	)

constructs a Radau grid with \( n+1 \) points on \( [a,b] \)

This may throw LinearAlgebraException.

Member Function Documentation

◆ differentiationMatrix()

virtual RealMatrix const & Kaskade::SDCGridImplementationBase::differentiationMatrix ( ) const

inlinevirtualinherited

Differentiation matrix \( D \).

This computes a matrix such that polynomials given by values at \( t_0, \dots,t_n\) can easily be differentiated.

The Lagrangian interpolation functions \( L_k \) are defined by \( L_k(t_i) = \delta_{ik} , \quad i=0,\dots,n\). The matrix \( D \in \mathbb{R}^{n+1\times n+1}\) contains the values

\[ D_{ik} = \dot L_k(t_i) \]

This way, if \( u \) is defined in terms of its function values \( v_i = u(t_i) \), its derivatives can be evaluated by a matrix-vector multiplication:

\[ \dot u(\tau_i) = (Dv)_i \]

Implements Kaskade::SDCTimeGrid.

Definition at line 183 of file sdc.hh.

◆ integrationMatrix()

virtual RealMatrix const & Kaskade::SDCGridImplementationBase::integrationMatrix ( ) const

inlinevirtualinherited

Integration matrix \( S \).

This computes a matrix such that functions given by values at \( t_0,\dots,t_n\) can be easily integrated.

Interpolation is based on the nodes \(t_0\dots,t_n\). Depending on the actual implementation, the initial point \( t_0 \) might be ignored (i.e. have a quadrature weight of 0).

The Lagrangian interpolation functions \( L_k \) are defined by \( L_k(t_i) = \delta_{ik},\quad i=1,\dots,n \). The matrix \( S \in \mathbb{R}^{n\times n+1}\) contains the values

\[ S_{ik} = \int_{\tau=t_i}^{t_{i+1}} L_k(\tau) \, d\tau \]

(the leading column being zero). This way, if \( u \) is defined in terms of its function values \( v_i = u(t_i),\quad i=1,\dots,n \), the integrals can be evaluated by a matrix-vector multiplication:

\[ \int_{\tau=t_i}^{t_{i+1}} u(\tau) \, d\tau = (Sv)_i \]

Implements Kaskade::SDCTimeGrid.

Definition at line 178 of file sdc.hh.

◆ interpolate()

virtual RealMatrix Kaskade::RadauTimeGrid::interpolate ( RealVector const & x ) const

virtual

Compute interpolation coefficients.

Returns a matrix \( w \in \mathbb{R}^{m+1\times n+1} \), such that the interpolation polynomial \( p \) to the values \( y_i \) at grid points \( t_i \) can be evaluated as

\[ p(x_i) = \sum_{j=0}^n w_{ij} y_j, \quad i=0,\dots,m. \]

Implements Kaskade::SDCTimeGrid.

◆ point()

double Kaskade::SDCTimeGrid::point ( int k ) const

inlineinherited

Time points in the time step.

This is a convenice function.

Definition at line 94 of file sdc.hh.

◆ points()

virtual RealVector const & Kaskade::SDCGridImplementationBase::points ( ) const

inlinevirtualinherited

Time points in the time step.

The time step \( [t_0, t_n] \) contains \( n+1 \) time points \( t_i \), including the end points. Those are provided here. The time points are stored in increasing order.

Implements Kaskade::SDCTimeGrid.

Definition at line 173 of file sdc.hh.