timeAdvance_template - File containing a class to deal the time advancing scheme More...

#include <TimeAdvance.hpp>

Inheritance diagram for TimeAdvance< feVectorType >:

Public Types
typedef feVectorType	feVector_Type

typedef ScalarVector	container_Type

typedef std::vector< feVector_Type >	feVectorContainer_Type

typedef std::vector< feVector_Type * >	feVectorContainerPtr_Type

typedef feVectorContainerPtr_Type::iterator	feVectorContainerPtrIterate_Type

typedef std::vector< std::shared_ptr< feVector_Type > >	feVectorSharedPtrContainer_Type

Constructor & Destructor
	TimeAdvance ()
	Empty Constructor. More...

virtual	~TimeAdvance ()
	Destructor. More...

Methods
virtual void	shiftRight (const feVector_Type &solution)=0
	Update the vectors of the previous time steps by shifting on the right. More...

Detailed Description

template<typename feVectorType = VectorEpetra>
class LifeV::TimeAdvance< feVectorType >

timeAdvance_template - File containing a class to deal the time advancing scheme

Author: Matteo Pozzoli matte.nosp@m.o1.p.nosp@m.ozzol.nosp@m.i@ma.nosp@m.il.po.nosp@m.limi.nosp@m..it

This class define an abstract method to build temporal discretization schemes. In particular we consider problems of the first order and the second order in time; that after space and time discretization, and suitable linearisation of non linear operator, we obtain a linear system to solve at each time step (or each iteration ):

$K U^{n+1} =F^{n+1}$

where $K$ is an opportune matrix, $U^{n+1}$ is the unknown vector and $F^{n+1}$ is the right hand side vector at the time $t^{n+1}$ . To determine $F^{n+1}$ we define the state vector $X^{n+1}$ that contained the informations about previous solutions.

First order problems:

$M \frac{\partial u}{\partial t} + L( u) = f ,$

where L is a generic nonlinear operator. We define U an approximation of u and the velocity vector an approximation of $\dot{u}$ at the time step as

$V^{n+1}=\frac{\alpha_0}{\Delta t} U^{n+1}- f_V^{n+1},$

where $\alpha_0$ is a suitable coefficient and is a linear combination of the previous solutions $X^{n+1}$ with coefficients $\alpha_i$ , that will be specified in the following.

Then the time discrete version is
$\frac{\alpha_0}{\Delta t}M U^{n+1} +A (U^{n+1})= f^{n+1}+ M f_V^{n+1}$

, that can be solved by any non-linear iterative solver (fixed point, Newton, ....). This class provides also a suitable extrapolation of $U^{n+1}$ , given by a linear combination of previous solutions and of order consistent with the time discretization formula.
In this part we want to extend the previous approach to second order problems in time.

Second order problems:

$\ddot{ u} + L( \dot{u}, u) = f$

where is non linear operator.

We consider the following semidiscrete problem

$M \frac{d^2 U}{d t^2} + D ( U, \frac{d U}{d t} ) + A ( U ) = f$

where is the mass matrix, the forcing term and is the vector of unknowns.

We define the following quantities

$V :=\frac{d U}{d t }\qquad W := \frac{d^2 U} {d t^2},$

where and are the velocity and the acceleration vectors, respectively.

At the time step , we consider the following approximations of and :

$V^{n+1}=\frac{\alpha_0}{\Delta t} U^{n+1}- f_V^{n+1},$

and

$W^{n+1}=\frac{\xi_0}{\Delta t^2} U^{n+1} - f_W^{n+1},$

where $f_V^{n+1}$ and $f_W^{n+1}$ are linear combinations of the previous solutions with suitable coefficients $\alpha_i$ and $\xi_i$ , respectively. If and depend on and we can linearise the problem using suitable extrapolations obtained by linear combinations of previous solutions with coefficients $\beta_i$ and $\beta_i^V$ respectively.