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

ALGORITHMSFOR SURFACEMOTIONPROBLEMS 7
defined on the grid jds =O, + 1, rt_2, . . . . As As -+ 0, the discrete initial data
converges to the flat line $ = 0 in the following fashion:
yo(j As) = 0 + O( (A#‘)
yb(j As) = 0 + O((As)“‘)
xo(j As) = x + O( As)
x;(j As) = 1 + O( As).
Iiowever, a marker particle numerical scheme (without regridding) blows up after
O(l/(ds)“*) time steps. With any relationship between Af and As of the form
At = (ds)J’, p> 4, the actual numerical blow up time is O(dt)O(l/(d~)“~) =
O(As)P- 112, which goes to zero as As vanishes, even though the real solution is finite
(in fact, zero in all derivatives) for all time.
APPENDIX B: GENERAL EN0 CONSTRUCTION
We build RM by induction on M. In each cell xj 8 x f xii r ~ R’ is defined as
follows:
R’(x; !P)=!P;+(x-xj)D+Y;
= Y;+(x-xj) Yyxj,xj+l],
where u[xj - p, . . . . sj + u] denotes the usual coefficient in the Newton interpoiating
polynomial. We also define kg,), = j, k,,, (l) = j+ 1. Suppose we have defined
R”-‘(x; F)forxj