1549301742-The_Theory_of_Difference_Schemes__Samarskii

522 Difference Methods for Solving Nonlinear Equations

approximation that the scheme a) offers in such matters more advantages

than the sche1ne b).

Practical implementations showed that the c0111putational procedures

of the scheme b) can work with a larger step in tin1e, thus reducing essen-

tially the total volume of computations and the time complexity despite

the extra iterations required in this connection.

However, a. preference relation between such schemes is some consen-

sus of opinion. The reader can encounter in the theory and practice various

schemes generating approximations of order 2 in tin1e and space:

As can readily be observed, they are not monotone, thus causing some

"ripple". This obstacle can be avoided by refining some suitable grids in

ti1ne. When solving equations of the form (13) with a weak quasilinearity

for the coefficients k = k(x, t), f = f(u) and c = c(x, t), con1mon practice

involves predictor-corrector schemes of accuracy 0( r^2 + h.^2 ). Such a

scheme for the choice c = k = 1, f = f( u) is available now:

y-y -

0. 5 7 = Yxx + f(y) ,
   ( 40)
   y-y 1 -.
   7 =^2 (Y.rr + Y.r,J + f(y),

whose composition is depicted in Figure 1.

y

y

y

X·

'·

Figure 1.

t J

We omit here theoretical investigations of the preceding schemes re-
lying on cumbersome calculations and leading to unsatisfactory and rough
estin1a.tes that can result in wrong reasoning. Such difficulties a.re, gener-
ally speaking, typical for nonlinear problen1s in tnany branches of scienu\