GenLinVcFV.hpp File Reference

Template class for the general vertex-centered finite-volume discretizations of linear PDEs. More...

#include <iostream>
#include "small_matrix.hpp"
#include "SAssemble.hpp"
#include "WTSScheme.hpp"

Include dependency graph for GenLinVcFV.hpp:

Classes
class	GenLinVcFV< T, Flux >
	Template class for the general vertex-centered finite-volume discretizations of linear PDEs. More...

---

Detailed Description

Template class for the general vertex-centered finite-volume discretizations of linear PDEs.

The implementation should accept arbitrary dimensions.

The class template depends on two parameters:

1. The user-defined class 'Flux' implementing the flux functions for the cell surfaces,
2. the type T, as in the class small_matrix<T>. The user-defined class 'Flux' should have the following member functions:
   • void PrintParam() should print the name and the parameters of the object.
   • void PreProcess(SparseMatrix<T> & A) should prepare the class for assembling the matrix 'A'.
   • void AssembleInnerFaceFlux(T factor, int face, MultiIndex & I, SparseMatrix<T> & A) should update the matrix 'A' by assembling the flux through one inner face, so that this flux is multiplied by 'factor'; this is the 'left' face of the control volume of the vertex with the index 'I' orthogonal to the direction of the dimension axis 'face'; the flux should be related to the unit face area w.r.t. the scaled index, because the grid steps are taken into account in 'factor'; NOTE that the function should update the entries in two matrix rows: those corresponding to both the control volumes intersecting over this face.
   • void AssembleBndFaceFlux(T factor, int face, MultiIndex & I, SparseMatrix<T> & A) should update the matrix 'A' by assembling the flux through a control volume boundary face, so that this flux is multiplied by 'factor'; this is the face orthogonal to the direction of the dimension axis 'face'; the user should distinguish between the two faces by the index 'I' the flux should be related to the unit face area w.r.t. the scaled index, because the grid steps are taken into account in 'factor'.
   • T det_J(MultiIndex & I, Array<int> & grid_sizes) - should return the determinant of the Jacobian of the transformation of the unit square to the problem domain; the value should correspond to the vertex with the index 'I'; 'grid_sizes' references the sizes of the grid.
   • void PostProcess(SparseMatrix<T> & A) should postprocess the matrix 'A' after assembling the control volume faces and the mass matrix; in particular, this function should assemble the matrix at the vertices where the Dirichlet boundary conditions are assumed (merely set the identity matrix at these vertices).
   • void AssembleDirichlet(MultiArray<T> & u) should assemble the Dirichlet values in the given vector 'u'. For better performance, these functions should be inline.

USAGE INSTRUCTIONS:

1. For the time-dependent problems:
   
   • If you have no sinks and sources, it is enough to specify the 'Flux' class and to declare an object of the class GenLinVcFV<T,Flux>. Then you can may use this object with the linear time-stepping schemes. But before you run the time-stepping scheme, you should initialize the Dirichlet boundary conditions in the initial condition using AssembleDirichlet (if you have any boundaries with such the conditions).
   • If you have sinks, sources or explicit terms, derive a new class from GenLinVcFV<T,Flux> and redefine the corresponding virtual functions. The use the new class instead of GenLinVcFV<T,Flux> as described above.
2. For the stationary problems:
   The member function 'Run' assembles the stiffnes matrix (without any sink etc terms) and the Dirichlet boundary conditions in the right-hand side and the initial conditions. To assemble further terms, derive a new class from GenLinVcFV<T,Flux>.

1. This class template accepts only GridVec s consisting of exactly one MultiVector and GridMatrices of exactly one SparseMatrix .
2. Nevertheless this template is capable of assembling discretizations for vector equations (up to now: with no reaction terms). For this, the functions in the user-defined class 'Flux' should update the matrices having the small blocks as the entries.
3. You can also use this template class for the structured grids on complicated domain geometries. The only assumption is that the user-defined flux functions are implemented for this.

Date:

Mar. 2, 2006 - created

---