48
3. Finite Difference Methods
In this case, the leading truncation error term for the CDS can be rewritten:
The leading error term of the first-order FDS or BDS schemes is:
When re is close to unity, the first-order truncation error of the CDS is
substantially smaller than the BDS error.
Now let us see what happens when the grid is refined. We consider two
possibilities: halving the spacing between two coarse grid points and inserting
new points so that the fine grid also has a constant ratio of spacings.
In the first case, the spacing is uniform around the new points, and the
expansion factor re a t the old points remains the same as on the coarse grid.
If the refinement is repeated several times, we obtain a grid which is uniform
everywhere except near the coarsest grid points. At this stage, a t all grid
points except those belonging to the coarsest grid, the spacing is uniform
and the leading error term in the CDS vanishes. After some refinements,
the number of points at which the spacing is non-uniform will be small.
Therefore, the global error will decrease just a bit more slowly than in a true
second-order scheme.
Grid 2h
i+ 1
Grid h
Fig. 3.3. Refinement of a non-uniform grid which expands by a constant factor re
In the second case the expansion factor of the fine grid is smaller than on
the coarse grid. Simple arithmetic shows that
r e , h = a ,
(3.23)
where h represents the refined grid and 2h, the coarse grid. Let us consider
a node common to both grids; the ratio of the leading truncation error term
at node i on the two grids is (see Eq. (3.22)):
The following relation holds between the mesh spacing on the two grids (see
Fig. 3.3):
3. Finite Difference Methods
In this case, the leading truncation error term for the CDS can be rewritten:
The leading error term of the first-order FDS or BDS schemes is:
When re is close to unity, the first-order truncation error of the CDS is
substantially smaller than the BDS error.
Now let us see what happens when the grid is refined. We consider two
possibilities: halving the spacing between two coarse grid points and inserting
new points so that the fine grid also has a constant ratio of spacings.
In the first case, the spacing is uniform around the new points, and the
expansion factor re a t the old points remains the same as on the coarse grid.
If the refinement is repeated several times, we obtain a grid which is uniform
everywhere except near the coarsest grid points. At this stage, a t all grid
points except those belonging to the coarsest grid, the spacing is uniform
and the leading error term in the CDS vanishes. After some refinements,
the number of points at which the spacing is non-uniform will be small.
Therefore, the global error will decrease just a bit more slowly than in a true
second-order scheme.
Grid 2h
i+ 1
Grid h
Fig. 3.3. Refinement of a non-uniform grid which expands by a constant factor re
In the second case the expansion factor of the fine grid is smaller than on
the coarse grid. Simple arithmetic shows that
r e , h = a ,
(3.23)
where h represents the refined grid and 2h, the coarse grid. Let us consider
a node common to both grids; the ratio of the leading truncation error term
at node i on the two grids is (see Eq. (3.22)):
The following relation holds between the mesh spacing on the two grids (see
Fig. 3.3):