FIFTH CANADIAN CONFERENCE ON NONDESTRUCTIVE ... - IAEA

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linear system of equations coupling the latest time level of the solution.
Marfurt agrees with this view in a recent review paper [7]. We, therefore,
restrict our attention to schemes which calculate the solution at the latest
time level explicitly.
The usual finite difference approximation to the second order form of the
equations is constructed by replacing the partial derivatives with second order
accurate central difference approximations. (The order of accuracy of a finite
difference approximation is the order of the leading term in its truncation
error. See [6], p 21). For example, the second partial time derivative of U is
approximated by
U(x,t) = -^ (U(x,t+At) - 2U(x,t) + U(x,t-At))
At
and the second partial space derivative of U by
U(x,t) = -i- (U(x+Ax,t) - 2U(x,t) + U(x-Ax,t))
xx Ax 2
Bond [5] gives a detailed description of this scheme.
There are a large number of approximation schemes for first order hyperbolic
systems and one must consider their accuracy, efficiency and reliability. The
numerical simulation developed by the author uses a "leap-frog" scheme proposed
by Kreiss and Öliger [8] which is accurate to fourth order in space and to
second order in time. These schemes are particularly useful because they can
be implemented efficiently, give good resolution of the solution and do not
dissipate energy. In addition, they have been used successfully in weather
prediction codes and have been the subject of extensive theoretical
investigation.
Detailed descriptions of the Kreiss and Öliger leap-frog scheme can be found in
[8] and [9], and so we give only a brief description here. The scheme is
constructed by approximating the time derivative of U by
U(x,t)t = Y^r (U(x,t+At) - U(x,t-At))
and the space derivatives by
U(x,t) = yi-^- |U(x-2Ax,t) - 8U(x-Ax,t) + 8U(x+Ax,t) - U(x+2Ax,t)J
The difference approximation of the space derivatives must be modified at
points adjacent to and on the boundaries of the space region to avoid the use
of points outside the region.