1549301742-The_Theory_of_Difference_Schemes__Samarskii

Heat conduction equation with constant coefficients 327

These boundary conditions generate an approximation of O(h^2 + T) for any

(} if we agree to consider p 1 = p 2 = 1 and

/()- = f(O t + r) + μ^1 (t + r)

r ' 0.5 h '

But the same procedure works for (} = ~ and the boundary conditions (67),

making it possible to achieve O(h^2 + r^2 ). Eventually, by successive use of

the members

hf32

P2 = l+ 3

zp = f(x, t + r) + ~~ (J"(x, t + r) + /(x, t + r)),

we arrive at scheme ( 64) of accuracy 0( h^4 + r^2 ) and the error of approxi-

mation O((h^4 + r^2 )/h) at the nodes x = 0 and x = 1.

5.2 ASYMPTOTIC STABILITY

1. p-stability. In studying various schemes for the heat conduction equa-
   tion we have already mentioned that the simplest way of relaxation of the
   stability condition connected with upper bounds on the step in time T is
   the transition to implicit schemes. All of the implicit schemes with a weight
   of the upper layer (} > () 0 , () 0 = ~ - ~ h^2 T-l, are stable. In particular, the
   schemes with (} = ~ and (} = 1 are absolutely stable. At the final stage
   of research the scientists are interested in the accuracy with which an ap-
   proxi1nate solution to a differential equation could be found. It is natural
   to try to reach a prescribed accuracy by carrying out the minimal possi-
   ble number of computer operations (arithmetic and logic), that is, with