Convergence and accuracy of homogeneous conservative schemes

The error of approximation in the class of smooth coefficients. The main 

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In Section 1.4 we have established conditions of the second-order local approximation for the conservative scheme (2) with linear nonnegative pattern functionals 4l[fe(s)] and i [/(s)] such as  

As can readily be observed, these conditions remain valid for any scheme from the primary family. 

In order to evaluate the order of accuracy for scheme (2), it is necessary to make the accurate account of the error z = y u being viewed as a solution of problem (3). Moreover, the desirable estimate should be expressed in terms of the right-hand side ip. In this direction the error of approximation to ip x) is considered first. If k x) and q x),f x) then 

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 

This means that scheme (2) provides a. local approximation of order 2, so that ip c Mh2, where M = const 0 is independent of h. 

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