50
2. BasicFinite-Difference Methods
the stability of amplifying schemes, which will be considered in Sections 2.3.2
and 2.5, concems the extent to which they can produce approximate solutions to
ordinary and partial differential equations that converge in the limit /)"t
/)"x
o.
0 and
2.3.2 Single-Stage Two-Level Schemes
The simplest techniques for the solution of the differential equation
d1/l
-
= F(1/I)
(2.32)
dt
are members of the general family of single-stage two-tirne-level schemes, which
may be written in the form
4Jn+1
4Jn
- - - - = aF(4Jn) + ßF(4Jn+l).
(2.33)
/)"t
Here 4Jn is the numerical approximation to 1/I(n/)"t) , and a + ß = 1 for consistency with the original equation. The preceding general form includes several
well-known elementary methods. When a = 1, ß = 0, the scheme is known
as forward differencing or Euler's method. Backward differencing corresponds to
the case a = 0, ß = 1, and the trapezoidal method is obtained when a = ß = !.
The finite-difference scheme (2.33) may be altematively expressed in the form
(2.34)
which is analogous to the integrated form of the original differential equation
1/1 [(n + 1)/)"t] = 1/I(nM) +
t: nÖl
F (1/I(t)) dt.
Although (2.33) and (2.34) are clearly equivalent, confusion sometimes arises in
determining the accuracy of numerical methods expressed in the integrated form
(2.34). If the numerical method is convergent of order r and if 1/1 is the solution
to the continuous problem, expansion of 1/1 in a Taylor series and substitution of
that series into the discrete derivative form (2.33) will yield a residual error of
O[(/)"tY), whereas substitution into the discrete integral form (2.34) will yield an
error of O[(My+ I ) . Of course, if the solutions to (2.33) and (2.34) converge to
the true solution as /)"t
0, they must converge at the same rate, because the two
schemes are algebraically equivalent. This rate of convergence is equal to the order of the global truncation error, which is the same order as the truncation error
associated with (2.33). The O[(My+l) truncation error associated with the integral form (2.34) is the local truncation error, or one-step error, and represents
the error introduced in each step of the integration, i.e., the difference between
1/In+1 as computed by one step ofthe finite-difference method and the exact solution to the differential equation at t = (n
2. BasicFinite-Difference Methods
the stability of amplifying schemes, which will be considered in Sections 2.3.2
and 2.5, concems the extent to which they can produce approximate solutions to
ordinary and partial differential equations that converge in the limit /)"t
/)"x
o.
0 and
2.3.2 Single-Stage Two-Level Schemes
The simplest techniques for the solution of the differential equation
d1/l
-
= F(1/I)
(2.32)
dt
are members of the general family of single-stage two-tirne-level schemes, which
may be written in the form
4Jn+1
4Jn
- - - - = aF(4Jn) + ßF(4Jn+l).
(2.33)
/)"t
Here 4Jn is the numerical approximation to 1/I(n/)"t) , and a + ß = 1 for consistency with the original equation. The preceding general form includes several
well-known elementary methods. When a = 1, ß = 0, the scheme is known
as forward differencing or Euler's method. Backward differencing corresponds to
the case a = 0, ß = 1, and the trapezoidal method is obtained when a = ß = !.
The finite-difference scheme (2.33) may be altematively expressed in the form
(2.34)
which is analogous to the integrated form of the original differential equation
1/1 [(n + 1)/)"t] = 1/I(nM) +
t: nÖl
F (1/I(t)) dt.
Although (2.33) and (2.34) are clearly equivalent, confusion sometimes arises in
determining the accuracy of numerical methods expressed in the integrated form
(2.34). If the numerical method is convergent of order r and if 1/1 is the solution
to the continuous problem, expansion of 1/1 in a Taylor series and substitution of
that series into the discrete derivative form (2.33) will yield a residual error of
O[(/)"tY), whereas substitution into the discrete integral form (2.34) will yield an
error of O[(My+ I ) . Of course, if the solutions to (2.33) and (2.34) converge to
the true solution as /)"t
0, they must converge at the same rate, because the two
schemes are algebraically equivalent. This rate of convergence is equal to the order of the global truncation error, which is the same order as the truncation error
associated with (2.33). The O[(My+l) truncation error associated with the integral form (2.34) is the local truncation error, or one-step error, and represents
the error introduced in each step of the integration, i.e., the difference between
1/In+1 as computed by one step ofthe finite-difference method and the exact solution to the differential equation at t = (n