Homogeneous difference schemes on non-equidistant grids

In order to develop a homogeneous conservative scheme on the non-equidistant grid O , we write down the balance equation 

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When the available functions k x), q x) and f[x) happen to be of the class 1] and their discontinuity points are known, a non-equidistant grid can be made so that all discontinuity points of the coefficients k, q and / would be nodal points of such a grid. We denote by any such 

In the case of continuous coefficients expressions (5) imply that Oj = ki i, di = qi and ipi = fi- If discontinuity points coincide with nodal points of the grid uij, that is, x = Xi j2, then the members a, di and Pi can be found in simplified form  

The error of approximation. We now investigate the error of approximation for scheme (4) on the non-equidistant grid u h by considering the equation for the error z = y — u  

On account of the balance equation (1) the error of approximation reduces to 

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Tikhonov, A. and Samarskii, A. (1962) Homogeneous difference schemes on non-equidistant grids. Zh. Vychisl. Mat. i Mat. Fiz., 2, 812-832 (in Russian) English transl, in USSR Comput. Mathem. and Mathem. Physics. 

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