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An algorithm for the numerical solution of systems of partial differential equations, by an explicit Finite Difference Scheme, using a nonregular space grid has been implemented.
English (United States)
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Algorithm for Solution of Partial Differential Equations by Means of a Finite
Difference Nonregular Scheme
An algorithm for the numerical solution of systems of partial differential
equations, by an explicit Finite Difference Scheme, using a nonregular space grid
has been implemented.
The basic idea, which makes this scheme different from others, is the
evaluation of the orthogonal derivatives, at each node of the grid, by means of
the derivatives computed following the directions defined by the node itself and
the surrounding ones.
As an example for a two-dimensional field the points P(1), P(2),...P(N) are
taken as surrounding the point P(O), in which the x and y derivatives have to be
computed; the algorithm takes the following form.
First of all the N triangles P(i)P(O)P(i+1) (i = 1....N and P(N+1) = P(1)) are
considered and the derivatives following the directions
Alpha(i) identical to P(i)P(O) and Alpha(i + 1) identical to P(i + 1)P(O), that is,
the values of
are used to evaluate the x and y derivatives by means of the well-known
relationships between the derivatives at a given point, along different directions.
Then the N values of the x and y derivatives, one for each triangle, are
averaged in order to obtain the final values at P(O).
Other ways could be used in order to obtain the orthogonal derivatives at
P(O) from the N directional derivatives, but the averaging has been chosen
because it leads to a very useful final expression. In fact it is easy to show that,
according to the described method, the following formulae are obtained: