1549301742-The_Theory_of_Difference_Schemes__Samarskii

212 Homogeneous Difference Schemes

The coefficients d( x ), IP( x) can be calculated by one and the same formula

1P(x) = F [p(x +sh), q(x +sh); f(x +sh)] ,

d(x) = F[p(x +sh), q(x +sh); q(x +sh)].

Pattern functionals are defined in the class of piecewise continuous

functions:

A[p(s), q(s)]

F[p(s), q(s); f(s)]

for p(s), q(s) E Q(o)[-1, OJ and

for p(s), q(s), f(s) E Q(^0 l[-1, 1].

It is easily seen from (17)-(18) that the exact scheme does not belong

to the family of schemes (16)-(17) in Section 2, whose pattern functionals

A[P(s)] and F[f(s)] depend solely on a single function. In the case of

equation (1) with constant coefficients p(x) =Po =canst and q(x) = q 0 =

canst the pattern functions cx(s, h) and {3(s, h) can be determined explicitly:

sh (x(l + s)h)

cx(s,h)=Po xh '

sh (x(l - s)h)

,6(s,h) =po xh '

x=VMo·

At the same ti1ne the coefficients a( x) and d( x) become constant values

a(x) =Po sh~:h)' d(x) = 2 qo th xh.

)( h 2

2. Schemes of arbitrary order accuracy. There is no difficulty to construct
   a scheme of arbitrary order accuracy by means of appropriate expressions
   for coefficients of an exact scheme.
   We now know from (16) that cx(s, h) and f3(s, h) are analytic functions
   of the parameter h^2 and, therefore, can be expanded in the series
   co co
   ( 19) cx(s, h) = L cxk(s) h^2 k, f3(s, h) = L {3k(s) h^2 k,

• k=O k=O
  where cxk(s) and {3k(s) are calculated by the recurrence formulae

u,(<) ~ j p(t) (} u,_,(,\jq(,) d,)

s

di, k > 0, cx 0 (s) = ;· p(i) di,

-1 -1 -1

1 1 1

fi,(,) ~IP(') (/ fi, , (,\)q(,\) d,\) di, k > 0, {3 0 ( s) = j p( i) di.

s