Geology Reference
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so that the difference model can be more conveniently written as
(U
i,n
- 2U
i,n-1
+ U
i,n-2
) = m
2
(U
i+1,n
-2U
i,n
+ U
i-1,n
)
(4.2.69)
or, as we will show, in the preferred form
U
i-1,n
- (2 + 1/m
2
) U
i,n
+ U
i+1,n
= (- 2U
i,n-1
+ U
i,n-2
)/m
2
(4.2.70)
The “recipe” inferred from Equation 4.2.70 is straightforward. For
convenience, let us assume that t = 0 begins with n = 3, and that
identical
static
solutions are available and assumed to hold at time steps n = 1 and 2. These two
solutions "start" our transient calculations from a state of rest. Now, the
transient solution will be computed one time level at a time, with n fixed,
beginning with n = 3. When a particular value of n is so “frozen,” the
specialized finite difference equation is written for each of the internal nodal
points i = 2, 3, 4, ... and so on, until i = i
max
- 1. The indexes i = 1 and i
max
are
specifically excluded from this process since their finite difference equations
require U values to the
left
of i = 1 and to the
right
of i = i
max
which do not exist.
At this point we have i
max
unknowns, but only i
max
- 2 linear equations; two
more equations, to be obtained from boundary conditions, are needed to
complete the formulation.
Consider a six-node system, for illustrative purposes, where the index i
varies from 1 to 6. Corresponding to i = 2, 3, 4 and 5, we have the equations
U
1,3
- (2 + 1/m
2
)U
2,3
+ U
3,3
= (-2U
2,2
+U
2,1
)/m
2
(4.2.71)
U
2,3
- (2 + 1/m
2
)U
3,3
+ U
4,3
= (-2U
3,2
+U
3,1
)/m
2
(4.2.72)
U
3,3
- (2 + 1/m
2
)U
4,3
+ U
5,3
= (-2U
4,2
+U
4,1
)/m
2
(4.2.73)
U
4,3
- (2 + 1/m
2
)U
5,3
+ U
6,3
= (-2U
5,2
+U
5,1
)/m
2
(4.2.74)
which provides four equations for the six unknowns to the left of the equal signs
(those on the right are known quantities from previous time steps).
As discussed, two additional equations are obtained from boundary
conditions. For example, the equation that would appear just before Equation
4.2.71 might take the form U
1,3
= 0 when modeling a fixed end, or U
1,3
- U
2,3
= 0 when assuming a free end, since u/ x = 0. The analogous equations that
would apply just after Equation 4.2.74 are U
6,3
= 0 and U
5,3
- U
6,3
= 0. Thus,
the final system of equations might take the form,
U
1,3
- U
2,3
= 0
(4.2.75a)
U
1,3
- (2 + 1/m
2
)U
2,3
+ U
3,3
= (-2U
2,2
+U
2,1
)/m
2
(4.2.75b)
U
2,3
- (2 + 1/m
2
)U
3,3
+ U
4,3
= (-2U
3,2
+U
3,1
)/m
2
(4.2.75c)
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