1549301742-The_Theory_of_Difference_Schemes__Samarskii

342 Difference Schemes with Constant Coefficients

The value yj+i on the relevant layer is to be determined by the recursion

j = 0, 1, ....

For the sake of simplicity we are working in a parallelepiped G such

that { 0 < x°' < l°', a= 1, 2, ... , p} and on a certain grid wh = {xi=

(i 1 h 1 , ... ,iphp), i°' = 0, 1, ... ,Na, h°' = l°'N;;^1 } equidistant in each of

the directions x °'. We refer to the (2p + 1 )-point operator A of second-order

approximation with the values

p

( 4) A y = L AO! y,

a=l

where y is the value of the function y(x, t) at a fixed node x = (i 1 h 1 , ... ,

ip hp), y(±la) = y(x(±la), t), x(±la) = (i1 h1 J ... , i0!-1 hO!-]J (iO! ± 1) hO!,

ia+i ha+i, ... , ip hp) is the node adjacent to x (x( +la l stands on the right

from x and x(-la) stands on the left from x). With the aid of the well-

established relations

I h^12 = h2 1 + h2 2 + ... + h2 p ,

we calculate the residual

provided the asymptotic expansion <pj = f(x, tj) + 0 (I h 12 + T) holds.
Scheme (3) is conditionally stable in the space C with respect to the
initial data, the right-hand side and the boundary data. The maximum
principle for the difference problem (3) may be of help in establishing the
indicated properties with further reference to the canonical form

p

+L ~ (yj(x(+Ia))+yj(x(-Ia)))+r<pj.

a=l °'