spectral time grid for defect correction methods with Lobatto points on \( [a,b]\) More...
#include <sdc.hh>
Detailed Description
spectral time grid for defect correction methods with Lobatto points on \( [a,b]\)
For Lobatto points, \( t_0 = a \) and \( t_n = b \) holds. Since \( t_0 \) is actually used as quadrature node, there are \( n+1 \) quadrature nodes in use. Thus, Lobatto quadrature as defined here is exact for polynomials of degree up to \( 2n-1 \).
Note that collocation with Lobatto points is A-stable but not L-stable, and therefore SDC methods on Lobatto grids may be a suboptimal choice for highly stiff problems (e.g., parabolic equations or Dirichlet b.c. realized as quadratic penalty). Consider using RadauTimeGrid in these cases.
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 | |
| LobattoTimeGrid (int n, double a, double b) | |
| constructs a Lobatto grid with \( n+1 \) points on \( [a,b] \). 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... | |
| 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... | |
| double | point (int k) const |
| Time points in the time step. More... | |
Member Typedef Documentation
◆ RealMatrix
|
inherited |
◆ RealVector
|
inherited |
Constructor & Destructor Documentation
◆ LobattoTimeGrid()
| Kaskade::LobattoTimeGrid::LobattoTimeGrid | ( | int | n, |
| double | a, | ||
| double | b | ||
| ) |
constructs a Lobatto grid with \( n+1 \) points on \( [a,b] \).
This may throw LinearAlgebraException.
Member Function Documentation
◆ differentiationMatrix()
|
inlinevirtual |
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.
◆ integrationMatrix()
|
inlinevirtual |
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.
◆ interpolate()
|
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()
|
inlineinherited |
◆ points()
|
inlinevirtual |
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.
◆ refine()
|
virtual |
perform refinement of the grid, filling the prolongation matrix
This may throw LinearAlgebraException.
Implements Kaskade::SDCTimeGrid.
The documentation for this class was generated from the following file:
