1549301742-The_Theory_of_Difference_Schemes__Samarskii

524 Difference Methods for Solving Nonlinear Equations

supposed that the temperature satisfies the equation

( 42)

au a au

cs(u) ot =ox (ks(ll) ax), s = 1,2.

At the sa1ne time, on the boundary of these phases the te1nperature i8 con-

stant and coincides with the temperature of the phase transition: u(x, t) =

tl*. The velocity of the boundary~ of the phase transition is subject to the

equation

Olli olll 8~

kl ox x=~+o - k^2 ox x=~-0 = ->. ot ,

if u < u* in the first phase and 'U > u* in the second one.

vVith the boundary condition for the phase transition tn v1ew, we

rewrite equation ( 42) by n1eans of the cl-function as

( c( u) + ). cl (u - 1l * ) ) Ou ot = ox a ( k( 'll) OU) ox ,

'll < 'll , u<1L,

'll > u , tl > u.

The method of smoothing is available for solving the Stephan problem.

As a tnatter of experience, this amounts to replacing the cl-function by a

nonzero cl-type function cl( u - tl*, 6.), not equal to zero only on the interval

(1l* - 6., u* + 6.) and must satisfy the norn1alization condition

1l * +"'

J cl(u - u*, 6.) du= 1.

tt*-~

The quasilinear equation

C(u) -. OU = -. a (-k(u,) -. Ou)

c)t c) .r c) J'

arises in the process of s1noothing the functions k 1 (u), k 2 ( tl), c 1 ( u) and c 2 ('u)
on the interval ( u - 6., u + 6.). All the schemes we have mentioned above
find a wide range of applications for its numerical solution. However, there
are other numerical methods for solving the Stephan problem concerned.