Abstracts for the Conference on Grid Adaptivity in Computational PDEs, Edinburgh, July 96	Next
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Optimization of the grid density for the method of characteristics

M Ziólko

Institute of Electronics AGH
ul.Czarnowiejska 78, 30-054 Kraków, Poland
ziolko@uci.agh.edu.pl

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Abstract

where are constant matrices. A property of the hyperbolic partial differential equations is that the eigenvalues of matrix are real. Let us assume that they are and .

It is well understood that the reversion of the time variable direction converts a stable system into an unstable system. The instability of the continuous final-boundary value problem leads to the numerical instability of the corresponding discrete system. The computer experiments showed that this instability is very strong. Large errors appeared which led to computing overflow. These difficulties make it impossible to solve backward the partial differential equation. Both boundary value problems (for and ) are unstable, however the instability is not as strong as the instability of final-boundary value problem. The computer calculations gave correct results for both boundary value problems. It supports the conclusion that stability is sufficient but not necessary condition for the accuracy of computer calculations. The computer modelling of processes occurring in practice pointed out that it is possible to use the method of characteristics in both directions of the space variable but only in one direction of the time variable. The opportunity of providing calculations in both space directions makes it possible to find experimentally the proper density of discretization. To start with let us assume the boundary condition

where the final time is sufficiently large (). Next let us assume the discretization and let us apply the method of characteristics to obtain the approximation of the boundary function

For the next step we can take this solution as a boundary condition for the backward in the space calculations. Once more we use the method of characteristics to obtain the approximation

By comparing (in the sense of a certain metrics) the assumed values (2) with the results of the computations (4) we obtain the error of the numerical method for the discretization . These calculations can be repeated for other values of to obtain information how the error and the time of computations depends on the discretization .

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