Higher-accuracy schemes

An exact scheme. For equation (1) in Section 1 it is possible to make the design of a homogeneous conservative exact three-point scheme so that a solution 2/j of the difference problem is identical with the exact solution u — u(x) of problem (1) from Section 1 at all the nodes of any grid 

Before giving further motivations, it will be convenient to set up problem (1) arising in Section 1 in the form 

It is worth noting here that the best scheme (14)-(15) of Section 2 is exact for g = / = 0. Indeed, the function 

We now turn to equation (1) on an equidistant grid d),. A key idea in the further development of an exact scheme is to express at any inner point (and, particular, at a = x ) of the interval (a j i, Xj+i) a solution u = u(x) to the second-order equation (1) in terms of the values U i, m,+i and the right-hand side f(x). This can be done using u(x) in the form 

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The accepted view is that the pattern functions u (xi) and vl2 x) will be declared to be solutions of the appropriate Cauchy problems 

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In this section we consider higher-accuracy schemes for the Dirichlet problem (1) of Section 1 in a rectangle. 

A higher-accuracy scheme. By minor changes we are led to a higher-accuracy scheme such as... 

Example 3 As a matter of experience, the intention is to use a higher-accuracy scheme similar to (12) being used for the heat conduction equation. As we have stated in Section 1, the factorized scheme... 

A higher-accuracy scheme in a rectangle. In Chapter 4, Section 5, the Dirichlet problem... 

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