โ Scribed by I.M. Gaisinskii
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under applicathat 148 when constructing the composite scheme from two schemes, the first may be any scheme of 2nd order of approximation, and the second, an approximation of the equation ilu iI'u at+'1a;.-~O with suitably chosen coefficient 'I. The method whereby the order of dispersion is improved by approximating a special equation (different from (1.1ยป, will be called the method of antidispersion. Expression (3.5) suggests a method of improving the order 6f approximation in schemes of odd order of approximation. It is sufficient to choose the meribers of the composite scheme in such a way that In the case of schemes of first order of approximation, this method is realized in /4/ the name of the antidiffusion method (see also /5/). We verified numerically that the tion of this method to third-order schemes in fact leads to the solution behaving like obtained by fourth-order schemes.
To sum up there are two procedures whereby the order of approximation can be increased: for schemes of even order we have to increase the order of dispersion, and for schemes of odd order, we have, to increase the order of dissipation. Given any ini tial scheme, by alternating these procedures, schemes wi th arbitrarily large order of approximation can be obtained. Then, on the basis of (1.15) , for infini tely differentiable initial data we can obtain an arbitrarily large rate of convergence. For initial data belonging to IV.~.
we can obtain a scheme whose rate of convergence in L,. is arbitrarily close to ct. The same rate of convergence can be obtained for so-called spectral methods /5/.
Notice in conclusion that similar results can be obtained for linear equations with variable coefficients, for systems of equations of hyperbolic type, and for equations of parabolic type. These results may also be extended to multi-dimensional equations.
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