8.5 Finite Difference Methods
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spacing in transformed space is arbitrary, but one usually takes Ati = 1).
Some authors claim that discretization becomes simpler, as the grid in the
transformed space appears simpler. This simplification is, however, only apparent: the flow does take place in a complex geometry and this fact cannot
be hidden by a clever coordinate transformation. Although the transformed
grid does look simpler than the non-transformed one, the information about
the complexity is contained in the metric coefficients. While the discretization on the uniform transformed mesh is simple and accurate, calculation of
the Jacobian and other geometric information is not trivial and introduces
additional discretization errors; i.e. it is here that the real difficulty has been
hidden.
The mesh spacing Ati need not be specified explicitly. The volume in
physical space, AR, is defined as:
If we multiply the whole equation by At1A[2A&, and replace J A t l A t 2 A t 3
everywhere by AR, then the mesh spacings Ati disappear in all terms. If
central differences are used to approximate the coefficients Pij, e.g. in 2D
(see Fig. 8.6):
the final discretized terms will involve only the differences in Cartesian coordinates between neighbor nodes and the volumes of imaginary cells around
each node. Therefore, all we need is to construct such non-overlapping cells
around each grid node and calculate their volume - the coordinates ti need
not be assigned any value and the coordinate transformation is hidden.
8.5.2 Method Based on Shape Functions
Although nobody seems to have tried it, the FD method can be applied to
arbitrary unstructured grids. One would have to prescribe a differentiable
shape function (probably a polynomial) which describes the variation of the
variable 4 in the vicinity of a particular grid point. The coefficients of the
polynomial would be obtained by fitting the shape function to the values of 4
at a number of surrounding nodes. There would be no need to transform any
term in the equation, as the shape function could be differentiated analytically to provide expressions for the first and second derivatives with respect
to Cartesian coordinates a t the central grid point in terms of the variable
values at surrounding nodes and geometrical parameters. The resulting coefficient matrix would be sparse but it would not have a diagonal structure
unless the grid is structured.
229
spacing in transformed space is arbitrary, but one usually takes Ati = 1).
Some authors claim that discretization becomes simpler, as the grid in the
transformed space appears simpler. This simplification is, however, only apparent: the flow does take place in a complex geometry and this fact cannot
be hidden by a clever coordinate transformation. Although the transformed
grid does look simpler than the non-transformed one, the information about
the complexity is contained in the metric coefficients. While the discretization on the uniform transformed mesh is simple and accurate, calculation of
the Jacobian and other geometric information is not trivial and introduces
additional discretization errors; i.e. it is here that the real difficulty has been
hidden.
The mesh spacing Ati need not be specified explicitly. The volume in
physical space, AR, is defined as:
If we multiply the whole equation by At1A[2A&, and replace J A t l A t 2 A t 3
everywhere by AR, then the mesh spacings Ati disappear in all terms. If
central differences are used to approximate the coefficients Pij, e.g. in 2D
(see Fig. 8.6):
the final discretized terms will involve only the differences in Cartesian coordinates between neighbor nodes and the volumes of imaginary cells around
each node. Therefore, all we need is to construct such non-overlapping cells
around each grid node and calculate their volume - the coordinates ti need
not be assigned any value and the coordinate transformation is hidden.
8.5.2 Method Based on Shape Functions
Although nobody seems to have tried it, the FD method can be applied to
arbitrary unstructured grids. One would have to prescribe a differentiable
shape function (probably a polynomial) which describes the variation of the
variable 4 in the vicinity of a particular grid point. The coefficients of the
polynomial would be obtained by fitting the shape function to the values of 4
at a number of surrounding nodes. There would be no need to transform any
term in the equation, as the shape function could be differentiated analytically to provide expressions for the first and second derivatives with respect
to Cartesian coordinates a t the central grid point in terms of the variable
values at surrounding nodes and geometrical parameters. The resulting coefficient matrix would be sparse but it would not have a diagonal structure
unless the grid is structured.