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Тезисы докладов

Пленарные доклады

In the approximation by point-wise techniques (such as finite-differences, spectral collocation or wavelet collocation) of functional equations, a substantial improvement, both from the point of view of the accuracy and the computational expenses, can be obtained by using two different grids to construct the discrete operator. One grid is the one used to {sl represent} te approximated solution, the other, called the {sl residual} or {sl collocation} grid, is where the equation is going to be satisfied. The two grids are related in such a way that the exact and the discrete operators have in common a space which is as large as possible, without increasing the number of degrees of freedom in the approximation. Such a property has been called {sl superconsistency} (see: D. Funaro, Superconsistent Discretizations, J. Scien. Comput., v. 17, 2002). In order to show that the same approach can be used with success also for time-dependent problems, we study finite-difference approximations of first-order scalar hyperbolic equation in one space dimension. The research has been developed in collaboration with G. Pontrelli from IAC-CNR, Rome. The representation grid is the usual uniform grid of width $Delta x$ and $Delta t$ in the space-time plane. The approximating equations are deduced from the discrete values of the solution, assumed to be computed over a classical six-points stencil of the representation grid, after collocation at a certain new point inside the same stencil. The position of the collocation point characterizes the approximation scheme, which now depends on two parameters, i.e. its coordinates, called $s$ and $r$. There will be actually three parameters after introducing another coefficient $nu$ related to numerical viscosity. For special choices of $s$, $r$ and $nu$, we will be able to obtain well-known schemes, but the interesting part is that infinite many other schemes, displaying an extended range of properties, can be also generated in this way. We discuss those that, according to our opinion, are particularly significant. For the linear transport equation we provide a general analysis of stability and consistency. In order to show that the idea can be adapted to more complicated problems, we give hints for the treatment of some nonlinear conservation laws.

Примечание. Тезисы докладов публикуются в авторской редакции

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