1549301742-The_Theory_of_Difference_Schemes__Samarskii

630 Economical Difference Schemes for Multidimensional Problems

where 1t = 'Uj and 1! = 11-j+l, Frmn here another conclusion can be drawn

that 1/' 1 = 0(1) and t/! 2 = 0(1), but

thereby clarifying a summarized approxin1ation provided by scheme ( 62),

Upon eliminating the intermediate value yj+l/^2 we have at our disposal the

factorized schen1e containing the values yj and y1 +l and approxin1ating the

heat conduction equation to 0( T + lhl^2 ) in the usual sense, But this schen1e

is stable only for com1nutative operators A1 and A2. No restrictions of this

sort is made for the additive scheme concerned.

14. LOS for a multidiinensional hyperbolic equation of second order. The
    1nethod of summarized approximation offers more advantages in design-
    ing absolutely stable and convergent locally one-dimensional schemes for
    equations of hyperbolic type. The object of investigation is the equation

(63)

where x = (:u 1 , .•• , J:p) _ERP and G is an arbitrary domain in the space RP

with the boundary r. G = G +rand

Qy = G x [O < t < T] , Qy = G x (0 < t < T].

The problem statement here consists of finding a continuous in the cylinder

Qy solution to equation (63) satisfying the boundary condition

(64) u = μ( x, t) for x E r, 0 < t < T,

and the initial conditions

(65) u(x, 0) = 1t 0 (x), ou(x, ot 0) = fi 0 (x) for xEG.

As usual, it is preassumed in a common setting that the problem
concerned is uniquely solvable and its solution lt = u(x, t) possesses all
necessary derivatives which do arise in all that follows. The domain of
interest G is still subject to the same conditions as we imposed in Section 5
for parabolic equations. Also, let w 7 = {t 1 =Jr, j = 0, l, ... } be a unifonu