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The notion of viscosity solutions was
developed first for conservation laws: first-order,
partial-differential equations linear, in the derivatives.
(Note that the eikonal equation is nonlinear in the derivatives.)
Conservation laws arise as simple models of inviscid fluid
flow,
and their weak solutions are made unique by a
viscosity principle as well. These viscosity weak solutions
are also called ``entropy'' or ``admissible'' solutions. An
excellent survey of this topic is in Lax (1973).
We note that
the connections between
viscosity, entropy, and
the uniqueness of weak solutions are completely understood
at present only in special situations, notably scalar
conservation laws in two variables.
We base our numerical work on the
observation that the gradient components of a solution
to the eikonal equation satisfy
conservation laws.
Moreover, the gradient components of
viscosity solutions of the eikonal equation are viscosity solutions
of the conservation laws.
As noted above, recent theoretical work (Crandall and Lions (1984))
and a lot of numerical ediffvidtimeence support the conjecture that the first-arrival
condition actually picks out a viscosity solution of the eikonal equation,
from amongst all the branches of traveltime (all of which solve the
eikonal equation locally).
Thus we can characterize
first-arrival-time fields by noting that
their gradient components are entropy or viscosity solutions
to the corresponding conservation laws.

** Next:** Upwind finite differences
** Up:** VISCOSITY SOLUTIONS AND UPWIND
** Previous:** Nonclassical and viscosity solutions
Stanford Exploration Project

1/13/1998