1549301742-The_Theory_of_Difference_Schemes__Samarskii

588 Econon1ical Difference Schemes for Multidimensional Problems

which are put together with a priori estimates for equation (59) with the

right-hand side

p

1/J = T2 Qpv = T2 L Ts-2Q~s)V' V =lit.

s=2

Two last summands in the preceding inequality are some quantities of

O(r^5 ), so there is some reason to make their contributions to the accurate

account of the accuracy z. Thus, scheme (58) converges in the grid space

0

TtV§ with the rate O(r^2 + lhl^2 ).

( 61)

The object of investigation is the system of hyperbolic equations

82 u

• 0

? = Lu+f(x,t), (x,t) E Qr,

i"

p

L = L Lcx(3,

a.f3=l

with the supplementary conditions

u = μ( x, t) for x E f , t E [0, T] ,

(62)

8u(x,O) _ ( f

u(x,O) = u 0 (x), Oi = U 0 X) or x E G,

under which it is required to find a continuous in the cylinder Qr solution.

Here the operator L = L~.f3=l Laf3 is specified by formula (L18). Observe

that a system arising from elasticity theory such as

82 u

• 0

? = f..l ~ u +(.A+μ) grad div u + f,

i"

where >. = const > 0 and μ = const are Lame 's coefficients, u = (u^1 , u",
... , uP) is a vector-function of order p, can be viewed as one particular case
of the system (61) for n = p and

fi. 'J = { 1'

u,

Here B is an arbitrary constant. In this context, several things are worth
noting. Condition (46) is automatically fulfilled. Condition (47) continues