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842 Chapter 19. Partial Differential Equations
t or n
x or j
staggered
leapfrog
Figure 19.1.5. Representation of the staggered leapfrog differencing scheme. Note that information
from two previous time slices is used in obtaining the desired point. This scheme is second-order
accurate in both space and time.
arrives at a given point after a time interval ∆t. In the first-order method, the
material always arrives from ∆x away. If v∆t ≪ ∆x (to insure accuracy), this can
cause a large error. One way of reducing this error is to interpolate u between j − 1
and j before transporting it. This gives effectively a second-order method. Various
schemes for second-order upwind differencing are discussed and compared in [2-3].
Second-Order Accuracy in Time
When using a method that is first-order accurate in time but second-order
accurate in space, one generally has to take v∆t significantly smaller than ∆x to
achieve desired accuracy, say, by at least a factor of 5. Thus the Courant condition
is not actually the limiting factor with such schemes in practice. However, there are
schemes that are second-order accurate in both space and time, and these can often be
pushed right to their stability limit, with correspondingly smaller computation times.
For example, the staggered leapfrog method for the conservation equation
(19.1.1) is defined as follows (Figure 19.1.5): Using the values of u n at time t n ,
compute the fluxes F n j . Then compute new values un+1 using the time-centered
values of the fluxes:
u n+1
j
− un−1 j
∆t
= −
∆x (F n j+1 − F n j−1) (19.1.30)
The name comes from the fact that the time levels in the time derivative term
“leapfrog” over the time levels in the space derivative term. The method requires
that u n−1 and u n be stored to compute u n+1 .
For our simple model equation (19.1.6), staggered leapfrog takes the form
u n+1
j
− un−1 j
v∆t
= −
∆x (unj+1 − u n j−1) (19.1.31)
The von Neumann stability analysis now gives a quadratic equation for ξ, rather than
a linear one, because of the occurrence of three consecutive powers of ξ when the
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