1549301742-The_Theory_of_Difference_Schemes__Samarskii

The sununarizecl approxin:1atio11 rnethocl 601

where t* > 0 is an arbitrary nun1ber. By resolving these equations we

obtain

yielding v 2 (t*) = u(t*).

Example 2 Recall now the statement of the Cauchy problem for the

transfer equation

(t=l,2,

-oo<x",<oo, t>O, u(x,O)=ft(x),

whose solution u(x, t) = p(x 1 - t, x 2 - t) is a "travelling wave" if p(x) is a

twice differentiable function. Since the operators L 1 and L2 are commuting,

it is plain to show that

u(x, t) = v( 2 )(x, t),

where v( 2 )(x, t*) is a solution of the system of equations

8v(l) 8v(l)

r +

0

=0, O<t<t*, v(l)(x,O)=p(x),

cN J: 1

8v( 2 ) 8v(2)

~~ + = 0 0 < t < t*, v( 2 )(x, 0) = v(l)(x, t*).

at ax2

Indeed, a solution to the first equation acquires the form

On the other hand, we find from the second equation that

i1nplying that v( 2 )(x, t) = ll(x, t).