Finite-Difference Method Definition/Meaning:
A widely applicable discretization method for the
solution of ordinary and partial differential equations. In this approach all
derivatives are replaced by approximations that involve solution values only, so
in general the differential equation is reduced to a system of
nonlinear
equations or linear algebraic equations. For example, in the problem
y² + by¢ + cy = d 0 ≤ x
≤ l,
y(0) = α, y(1) =
β
where b, c, d, α, and ft are given constants, the interval [0,1] is first
divided into equal subintervals of length h; h is called the stepsize (or mesh
or grid size). This gives the mesh points (or grid points) xn,
xn = nh,
n = 0,1, ... N + l,
h = 1/(N + 1)
At interior mesh points the derivatives are now replaced by finite-difference
approximations, e.g.
y¢(xn) ~ (1/2h)[y(xn
+ 1) - y(xn - 1)]
y² (xn)
~ (1/h2)[y(xn + 1)
- 2y(xn) + y(xn
- 1)]
When combined with the boundary conditions these approximations result in a
system of equations for approximations to y(xn), n = 1, 2, . . . , N. Nonlinear
differential equations yield a system of nonlinear equations.
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