2.2 Stability and Convergence
39
the solution of the continuous equation fails to satisfy the finite-difference fonnulation. Under the assumption that 1{! is sufficiently smooth, its value at adjacent
grid points can be obtained from a Taylor series expansion about (nßt, j ßX) and
substituted into (2.11) to yield
(2.12)
The right side of (2.12) is the truncation error of the finite-difference scheme. The
order of accuracy of the scheme is detennined by the lowest powers of ßt and
ßx appearing in the truncation error. According to (2.12), the upstream scheme
is first-order accurate in space and time. If the truncation error of the finitedifference scheme approaches zero as ßt -+ °and S» -+ 0, the scheme is
consistent. Inspection of (2.12) clearly shows that the upstream scheme is consistent. Although it is not difficult to design consistent difference schemes, this
property should not be taken fOT granted. One sometimes encounters methods
that require additional relations between M and ßX, such as ßt / Sx -+ 0, in
order to achieve consistency.
2.2 Stability and Convergence
The preceding measures of accuracy do not describe the difference between the
numerical solution cP' J and the true solution 1{!(nßt, j ßX) , which, of course, is the
most direct measure of the quality of the numerical solution . Before discussing
this error one needs a way to measure its size, i.e., one needs to define a norm.
The general mathematical notation for a nonn is a pair of vertical bars 11 11. In thc
following we will be concemed with the maximum nonn and the Euclidean, or
l2, nonn. The maximum norm, defined as
(2.13)
is simply the extremum of the grid-point values. The Euclidean, or l2, nonn is
defined as
(2.14)
If the constant scaling factor ßx is ignored, (2.14) is just the length of an N -
dimensional vector (hence the name Euclidean nonn). The inclusion of the ßx
factor makes (2.14) a numerical approximation to the square root of the spatial
integral of the function times its complex conjugate, cPcP*, whence the name l2
39
the solution of the continuous equation fails to satisfy the finite-difference fonnulation. Under the assumption that 1{! is sufficiently smooth, its value at adjacent
grid points can be obtained from a Taylor series expansion about (nßt, j ßX) and
substituted into (2.11) to yield
(2.12)
The right side of (2.12) is the truncation error of the finite-difference scheme. The
order of accuracy of the scheme is detennined by the lowest powers of ßt and
ßx appearing in the truncation error. According to (2.12), the upstream scheme
is first-order accurate in space and time. If the truncation error of the finitedifference scheme approaches zero as ßt -+ °and S» -+ 0, the scheme is
consistent. Inspection of (2.12) clearly shows that the upstream scheme is consistent. Although it is not difficult to design consistent difference schemes, this
property should not be taken fOT granted. One sometimes encounters methods
that require additional relations between M and ßX, such as ßt / Sx -+ 0, in
order to achieve consistency.
2.2 Stability and Convergence
The preceding measures of accuracy do not describe the difference between the
numerical solution cP' J and the true solution 1{!(nßt, j ßX) , which, of course, is the
most direct measure of the quality of the numerical solution . Before discussing
this error one needs a way to measure its size, i.e., one needs to define a norm.
The general mathematical notation for a nonn is a pair of vertical bars 11 11. In thc
following we will be concemed with the maximum nonn and the Euclidean, or
l2, nonn. The maximum norm, defined as
(2.13)
is simply the extremum of the grid-point values. The Euclidean, or l2, nonn is
defined as
(2.14)
If the constant scaling factor ßx is ignored, (2.14) is just the length of an N -
dimensional vector (hence the name Euclidean nonn). The inclusion of the ßx
factor makes (2.14) a numerical approximation to the square root of the spatial
integral of the function times its complex conjugate, cPcP*, whence the name l2