Finite Difference Representation
The ultimate goal for the solution of a PDE over a continuous domain is to reduce it to
discrete model which are suitable for high speed computers. One of the standard approach is
using Finite Difference Methods which are applicable to all type of
second order PDE.The main
steps in this approach are as follows:
1. Discretization
2.
Replacing the derivatives by FD representation
3.
Solving the corresponding Difference equations to get the numerical values of ‘u’ at the
discretized nodes
The first and third point will be discussed later. Let us discuss the approximation of the partial
derivatives by corresponding Finite Differences.
Let u(x, y) be the dependent variable .The Taylor’s series expansion can be written as:
2
2
2
)
,
(
2
)
(
)
,
(
)
,
(
)
,
(
x
y
x
u
x
x
y
x
u
x
y
x
u
y
x
x
u
Retaining
terms upto first order,we have,
)
(
)
,
(
)
,
(
x
O
x
y
x
u
y
x
x
u
x
u
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 Spring '14
 Prof.RamaBhargava
 Differential Equations, Numerical Analysis, Equations, Partial Differential Equations, truncation error, finite difference, Finite difference method, Finite differences, central difference formula

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