1549301742-The_Theory_of_Difference_Schemes__Samarskii

Conservative schemes 159

provided conditions (20)-(21) hold.

In what follows we deal everywhere with the primary family of ho-

rnogeneous conservative schen1es (16), (17) and (16^1 ), (17) as well as with

linear nonnegative pattern functionals A[k(s)] and F[f(s)] still subject to

conditions (20) and (21) of second-order approxi111ation.

In the sequel scheme (17), (15) will be called the best possible.

3.3 CONVERGENCE AND ACCURACY OF HOMOGENEOUS

CONSERVATIVE SCHEMES

1. The error of approximation in the class of smooth coefficients. The main
   point of the theory is the accurate account of the accuracy of the uniform
   scheme (16)-(17) in the class of continuous and discontinuous functions
   k(x), q(x) and f(x). In preparation for this, let u(x) be an exact solution
   of the original problem

(k(x) u^1 (x))

1

• q(x) u(x) = -f(x), O<x<l,
  (1)
  u(O)=u 1 , u(l)=u 2 , k(x)2c 1 >0, q(x)>O,

and let y = y( x) be a solution of the difference problem

(2)

(ayi:) x - d(x) y = -<p(x), x = ih , i = 1, 2, ... , N - 1 ,

a(x) 2 c 1 > 0, d(x)>O,

which belongs to the primary family of conservative schemes designed in
Section 2.4.
Common practice involves the error z(x) = y(x) - u(x) being a grid
function for x E wh. Inserting the representation y(x) = z(x) + u(x) in (2)
and assuming u(x)' to be a known function, we may set up the difference
problem for the error

(3)

Az= (az 10 )x-dz=-1f;(x),

z(O) = z(l) = 0,

where the residual

x = ih , i = l, 2, ... , N - 1 ,

Cl 2 c 1 > 0,

(4) 1/;(x) =Au+ <p(x) = (au,r:)x - cfo + <p