10. Numerical Modelling
319
is refined. And. it is convergence if the discretized équations of the given
different m équation sobres towards the exact solution as the grid spacing reduces. However. the numetical scheme for utilizing the discretization
sfaould uct tirensry tie errons during the process towards the solution and
ff dictâtes the scabui~. ne— u me ratto of the error lu the ith grid point at
n— i — ^tsç
mm at tue ".m steç *,s suould be less than or equal
ro one. By -ammrrnp rmeenaatiam laws. the mtmerfoal scheme is usually
been framed from m PB Es wmch in mm. hwe be» buüt on t he corservation laws. However. the -errors due t© coarse grids in general are due to
non-conservation but these are difficult to quantify
10.4.1 Discretization techniques
10.4.1.1 Finite différence method (FDM)
Finite différence method involves approximation of the governing équations
in differential form with différence équations obtained by replacement of
the dérivatives with finite. algebraic différences quotients. The domain is
first divided into grid, followed by the approximations of the continuons
fonctions at finite nodal values, i.e., one algebraic équation per degree of
freedom at a given grid node. Taylor sériés approximation is the most sought
after method of discretization, especially for structured grids. Further the
linear System of simultaneous algebraic équations is solved to obtain the
values at the grid nodes only.
10.4.1.2 Finite volume method FVM)
Finite volume method cornes into play when an intégral représentation of
the govFTTiwarin-n^ rar be formed. The problem domain is discret i/.od
into a finwp number of control volume oonnected together and the ronseï
vation laws are etivroj t© eam conirol volume. Here, the Gauss theorems
are employée to couvert the m'-np intégrais. The node of the grid is gen
erally Jocated a* the centre rd cf mm of the control volume. Many computiitiona. nuîd «tyna.Tr~.rs; pacages — -. <- use of this method as it has the added
advanrage of
géométries.
10.4.1.3 Finite element method (FEM)
The finite element method formulâtes the solution to a problem by subdi
tiding the domain into a finite number of éléments. The governing équations
319
is refined. And. it is convergence if the discretized équations of the given
different m équation sobres towards the exact solution as the grid spacing reduces. However. the numetical scheme for utilizing the discretization
sfaould uct tirensry tie errons during the process towards the solution and
ff dictâtes the scabui~. ne— u me ratto of the error lu the ith grid point at
n— i — ^tsç
mm at tue ".m steç *,s suould be less than or equal
ro one. By -ammrrnp rmeenaatiam laws. the mtmerfoal scheme is usually
been framed from m PB Es wmch in mm. hwe be» buüt on t he corservation laws. However. the -errors due t© coarse grids in general are due to
non-conservation but these are difficult to quantify
10.4.1 Discretization techniques
10.4.1.1 Finite différence method (FDM)
Finite différence method involves approximation of the governing équations
in differential form with différence équations obtained by replacement of
the dérivatives with finite. algebraic différences quotients. The domain is
first divided into grid, followed by the approximations of the continuons
fonctions at finite nodal values, i.e., one algebraic équation per degree of
freedom at a given grid node. Taylor sériés approximation is the most sought
after method of discretization, especially for structured grids. Further the
linear System of simultaneous algebraic équations is solved to obtain the
values at the grid nodes only.
10.4.1.2 Finite volume method FVM)
Finite volume method cornes into play when an intégral représentation of
the govFTTiwarin-n^ rar be formed. The problem domain is discret i/.od
into a finwp number of control volume oonnected together and the ronseï
vation laws are etivroj t© eam conirol volume. Here, the Gauss theorems
are employée to couvert the m'-np intégrais. The node of the grid is gen
erally Jocated a* the centre rd cf mm of the control volume. Many computiitiona. nuîd «tyna.Tr~.rs; pacages — -. <- use of this method as it has the added
advanrage of
géométries.
10.4.1.3 Finite element method (FEM)
The finite element method formulâtes the solution to a problem by subdi
tiding the domain into a finite number of éléments. The governing équations