曲率依赖速度下的前沿演化算法：1988年Osher的Hamilton-Jacobi形式

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"这篇文章是Stanley Osher在1988年发表的，标题为《Fronts Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations》。该研究主要关注利用Hamilton-Jacobi公式来解决曲率依赖速度的前沿推进问题，提出了一种新的数值算法——PSC算法，可以处理复杂的前沿动态，如被底层流体被动输送、尖锐梯度和尖点的形成，以及拓扑结构的合并和断裂。" 本文详细阐述了如何利用Hamilton-Jacobi（哈密顿-雅可比）形式化方法来开发新算法，这些算法旨在跟踪那些速度与曲率有关的前沿运动。曲率依赖速度可能是一个任意的曲率函数，且前沿也可以由底层流动被动推动。作者提出的方法，即PSC算法，能够近似求解类似于带有抛物型右端项的哈密顿-雅可比方程。为了求解这些方程，文章采用了源自双曲守恒律的非振荡方案，这些方案具有不同阶的精度，能够精确捕捉移动前沿中出现的尖锐梯度和尖点。PSC算法的一个关键优点是它自然地处理了前沿的拓扑变化，如前沿的合并与分裂。在数值模拟中，这样的拓扑变化通常是个挑战，而PSC算法提供了一种有效应对这些问题的方法。文章详细讨论了算法的实现细节，包括如何避免振荡和保持数值稳定性，这对于处理这类问题至关重要。此外，Osher和Sethian的工作还探讨了算法的效率和适用性，可能对计算物理学、流体力学、图像处理以及任何涉及前沿运动和形状演化的领域产生深远影响。通过这些算法，科学家和工程师能更好地理解和模拟物理现象，例如燃烧、侵蚀、界面动力学等。 "level-set. Osher 1988." 提供了一个基于Hamilton-Jacobi理论的前沿追踪算法，对于理解和解决曲率依赖速度的前沿运动问题提供了强大的工具，其贡献在于数值方法的创新和对复杂几何变形的有效处理。

OSHER AND SETHIAN

are not unique, and an additional entropy condition is needed to select the correct

viscosity limit. In order to assure that the solution to Eq. (2.7) be the unique limit

as E -+ 0 of Eq. (2.6), any of an equivalent class of entropy conditions is imposed

[17, 21, 281. The relevant one for our purposes is geometric, namely that charac-

teristics flow into a shock in the direction of increasing time. This means, for a

piecewise continuous weak solution U(X, t) having a jump moving with

--$ = S(t) =

G(u,) - G(ur)

u[-u, ’

that G’(u,) > S> G’(u,).

For the moving curve problem Eq. (2.1), or equivalently, Eq. (2.2), with

1, we

need an entropy condition which yields the unique limit solution as E -+ 0, of the

problem with

= 1 - EK. Imagine the curve y as a flame separating a burnt region

on the inside from an unburnt region on the outside, and an indicator function for

the burnt region was defined to be 4(x, y, t) = 1 if the particle at (x, v) is burnt at

time

and zero otherwise. In [31] the following entropy condition was suggested:

if 4(x, y,

t*)

= 1, then 4(x, y,

= 1 for

t*;

i.e., once a particle is burnt it remains

burnt. This was shown to be equivalent to requiring that ignition curves flow into

corners. In [32] numerical evidence was provided to show that the weak solution

generated by this entropy condition is indeed the correct limiting solution. We now

prove that this is so.

We consider a small section of the curve

=f(x, y), which, without loss of

generality, we can write as y = Y(x,

t).

We insert this into the expression

fz + f; =

1/F2,

arriving at

Y = (1 + Y2)1’2

(

E yx,

1 + (1 + Yi)3’2 .

)

Here, we have also chosen a positive square root. Letting U= Y, and taking the x

derivative of the above, we have [33]

(2.9)

for G(u) = - (1 + u2)l/‘, G(u) concave. The criterion for the inviscid limit problem

given in [31] is easily seen to be that characteristics propagate into shocks for

Eq. (2.9) with E = 0, that is, into

corners

for Eq. (2.8) with

F =

1. For concave G(u),

this is well known to be equivalent to the statement that limits of solutions to

Eq. (2.9) (and thus Eq. (2.8)) converge to solutions satisfying this criterion [21].

We may rewrite Eq. (2.8) in the following form, namely

[

E yxx

yt-- l+(l+yy

(C1+y31’2)=0

ALGORITHMS FOR SURFACE MOTION PROBLEMS 1

which is a Hamilton-Jacobi equation with second-order viscosity. For the

ore

general case of a speed function F(K), we have

(2.10)

which is also a Hamilton-Jacobi equation with second-order perturbation if the

speed function P satisfies (1) F(0) # 0 and (2) F’(O) # 0.

This formulation can be used to devise a numerical algorithm to approximate the

solution of a curve propagating with curvature-dependent speed, as long as the

remains a function, using the advanced technology for shock

ever, there is a different formulation of the problem which yields

Hamilton-Jacobi-type equation and does not require that the propagating from

remain a function. Define a Lipschitz continuous function 6(x, y, t) so that at t= 0,

4(x, y, 0) > 1 inside the burnt region Q, i.e., the region bounded by l’(O),

4(x, y, 0) < 1 outside Q, and 4(.x, y, 0) = 1 on &A Next, let 4(x, y, t) be defined by

4(x, Y, t) = C,

ere t =f(x, y) is defined from Eq. (2.2) for any fixed constant C.

This yields

and hence F*(Cz + 4;) = I$:. Thus, choosing the direction of propagation to be out-

wards, we have

- F(K)(qv + p)l’* = 0,

(2.11)

where

This is also a Hamilton-Jacobi equation with second-order right-hand s

However, this different formulation allows us to compue the solution even when

front is not a function and when two “burnt” regions merge together.

Using this formulation, and the recent theory of viscosity solutions to

amilton-Jacobi equations, Barles [l ] has proven that the entropy condition in

[3 1 ] picks out the unique viscosity solution even when the front is not a fn~~tion.

He defines 4(x, y, 0) = (I- d(x, y; a))+ + d(x, y; O”), where x’ = max(x, 0)

DC is the complement of 9, and evolves 4 according to Eq. (2.11) in the special

case

P= 1. He then chooses the unique viscosity solution which is characterized by the

entropy condition of Crandall-Lions [5] and shows that the resulting surface

y(t) = K?,, defined by 892, = (x, y, 14(x, y, t) = I), evolves according to the entropy

condition in [31].

Our results easily extend to initial surfaces. Suppose the surface v(

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曲率依赖速度下的前沿演化算法：1988年Osher的Hamilton-Jacobi形式

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