Fronts Propagating with Curvature- Dependent Speed: Algorithms ...

ALGORITHMS FOR SURFACE MOTION PROBLEMS 25
where TV is the total variation of a BV function as defined in [ 171. To continue the
algorithm, we then solve the initial value problem (Eq. (3.1)), with ‘Y(x, 0) =
R”(x; Y’), either exactly (as in the Godunov scheme) or approximately, using any
other monotone approximation. To obtain the new !P+ ‘, we define
where d t is taken small enough so that only waves from adjacent cells interact.
We simplify this method for our calculations. First, instead of solving the exact
problem, we approximate the Godunov flux by the simpler monotone flux g,,. The
numerical time integration can be performed either by formally replacing higher
time derivatives to arbitrary order by space derivatives, see [ 131, or by producing a
non-oscillatory Runge-Kutta type algorithm [ 29 ] from the semi-discrete for-
mulation
$ R”(x,: ; Y), $ R”(x; ; !F)
We note that for first-order monotone approximations to a linear equation
U, = -u,, the Hamilton-Jacobi and conservation law formulations yield the same
schemes. However, differences occur for higher order methods. A second-order
approximation for conservation form gives
and
AX
Tm[D-D-Yj,D-D,Yj~
for Hamilton-Jacobi form (m is defined below). The first is only first-order accurate
near critical points because the term with m[ ] gives u, + Q(dx), and the coef-
ficient in the O(dx) is not Lipschitz continuous near these points. The second is
uniformly second-order accurate since the analogous term yields YXX + Q(Ax).
also have estimates
B. Several Dimensions
In general, the Hamilton-Jacobi equations in several dimensions cannot be recast
as a simple system of conservation laws. However, using information gleaned from
the one-dimensional correspondence and resulting scheme design, we may now con-