1549301742-The_Theory_of_Difference_Schemes__Samarskii

376 Difference Schemes with Constant Coefficients

The identity (37) implies the stability with respect to the initial data

in the norm (40): 11yj+i11. = 11 y^0 II. for all j = 0, 1, ....

So, condition (39) is sufficient for the stability of scheme (32) with

respect to the initial data in the norm ( 40). In particular, scheme (32) with

O" = 0 is stable with respect to the initial data under the condition

(41) T < h.

Often this stability condition is named the Courant condition because

it has been proved for the first time by R. Courant, C. Friedrichs and G.

Levy in 1928.

For the equation in the dimensional variables

( 42)

condition ( 41) takes the form T < h/ a, where a is the sound velocity.

4. Determination of nonsmooth solutions by the difference method. Nu-
   merous problems of mathematical physics describing shock processes in
   gases, liquids and solids lead to the problem of determining nonsmooth
   solutions of second-order hyperbolic equations, the simplest of which is
   equation ( 42) of vibrations of a string. Since those solutions do not pos-
   sess the second-order derivatives involved in the equation, the words "a
   solution satisfies the equation" should be understood in some generalized
   sense. One of the possible definitions of generalized solutions is clue to the
   fact that the differential equation follows from an integral conservation law
   if continuous derivatives emerged in equation ( 42) exist. In this case the
   generalized solution is meant as a function u( x, t) having in the domain
   G = {O < x < l, 0 < t < T} the bounded piecewise continuous derivatives
   au/ ax, au/ at and satisfying the integral equation

f (

au 2 au )

8t dx + a ox dt = 0 ,

c

where C is an arbitrary closed curve in the domain G. If the first derivatives
are discontinuous, then for the characteristics x ± at = canst the jump
conditions should be satisfied:

[au at J _ - ±a [au ax J , [.f] :=.f(~+O)-.f(~-0) as ~=x±at.