Numerical Simulation for Blood Flow - Microsurgery: Advances, Simulations, and Applications - page 173

difference (FD) discretization methods, the grid is usually locally
structured, i.e., each grid node may be considered the origin of a
local coordinate system, whose axes coincide with grid lines. This
also implies that two grid lines belonging to same families, say Y 1 , do
not intersect, and that any part of grid lines belonging to different
families, say Y 1 = const. and Y 2 = const., intersect only once. In three
dimensions, three grid lines intersect at each node; the nodes of
three lines intersect each other at any other point. Each node is
uniquely identiied by a set of indices, which are the indices of the
grid lines that intersect at it, ( i , j ) in 2D and ( i , j , k ) in 3D. The neighbor
nodes are deined by increasing or reducing one of the indices by
unity. The generic scalar conservation equation is differential form.
As it is linear in G , it will be approximated by a system of linear
algebraic equations, in which the variable values at the grid nodes
are the unknown values. The solution to this system approximates
the solution to the PDE. Each node thus has one unknown variable
value associated with it and must provide one algebraic equation.
The latter is a relation between the variable value at that node and
those at some of the neighboring nodes. It is obtained by replacing
each term of the PDE at the particular node by a inite-difference
approximation. Of course, the numbers of equation and unknown
must be equal. At boundary nodes where variable value are given
(Dirchlet conditions), no equation is needed.
When the boundary conditions involve derivatives (as in
Neumann conditions), the boundary condition must be discretized
to contribute an equation to the set that must be solved. The idea
behind inite-difference approximations is borrowed directly from
the deinition of a derivative:
A geometrical interpolation is shown in Fig. 5.9 to which we
shall refer frequently. 2 The irst derivative ∂ G /∂x at a point is the
slope of the tangent to the curve f ( x ) at that point, the line marked
“exact” in the igure. Its slope can be approximated by the slope of
a line passing through two nearby points on the curve. The dotted
line shows approximation by the slope of a line passing through two
nearby points on the curve. The dotted line shows approximation
by a forward difference; the derivative at x i is approximated by
the slope of a line passing through the point x i and another point
at x i + δx . The dashed line illustrates approximation by backward
difference: for which the second point is x i

δx . The line labeled
“central” represents approximation by a central difference: It uses

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