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Upwind finite differences
Conservation laws are commonly solved by
upwind finite differencing, a numerical technique
that plays an important role
in modern computational fluid dynamics; see for
example Harten et al. (1987)
for a review.
Depending on the flow direction, upwind finite-difference schemes use a
backward or forward finite-difference approximation to the derivative operator.
The sense of the derivative is switched so that only field points
from upstream in the flow are used to update any given point.
This direction switching is forced by the appearance of discontinuities
(shocks) in the flow variables.
For the eikonal equation, the gradient components of traveltime
can be discontinuous: rays, whose direction is determined by the gradient
components, ``break'' at interfaces in the model where velocity contrasts
occur.
Reshef and Kosloff (1986) observe stability problems
in a centered finite-difference scheme at these interfaces.
Upwind finite-difference methods, on the other hand, are stable because
they mimic the underlying physics of the problem in two important aspects.
First, they add numerical viscosity to the equations, thus finding
a smooth viscosity solution. Second, they copy the behavior of
continuum flow by taking their information from upstream.
Careful inspection of Vidale's ordering of gridpoints reveals that
he applies the basic ``box'' (four-point) difference formula only
in the upwind sense (see Vidale, 1988, Figure 5), having used an
alternative extrapolation at
the few points at which the flow (i.e. ray field)
changes direction along the
computational front. Those points are local traveltime minima. His scheme
is thus an upwind finite-difference method, although not
presented as such. It is an implicit method, as
it connects more than one value on the grid level being updated.
The ordering of points is required to achieve a closed-form
solution of the difference formulas, as opposed to an iterative
approximate solution (as is often chosen with implicit schemes).
Our formulas are explicit, in contrast, and do not
require any special ordering of points.
Next: EIKONAL EQUATION
Up: VISCOSITY SOLUTIONS AND UPWIND
Previous: Conservation laws
Stanford Exploration Project
1/13/1998
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