# The Finite Differences Method

The method consists in replacing each derivative in the equation by a discretization (usually truncated Taylor series). There are a lot of schemes, depending on the chosen discretization for each derivative. After the discretization, we can obtain explicit schemes - if there's no need to solve a system of equations, just to walk the grid nodes - or implicit if we have to solve a system of equations for each row of the grid.
Domain discretization have to be accomplished by means of quadrilaterals parallel to the X and Y axis. Usually quadrilaterals are of equal size.
If a finite differences scheme needs information of the n row to compute the n+1 row, it is called one step scheme. Those that need information about several rows, are called multi-step. A multi-step scheme using m steps needs the solution values in the first (m-1) levels, or they must be calculated using other method.
If the approximate solution that a method obtain converge to the true equation solution when the mesh spacing tends to zero, the scheme is convergent. A scheme is stable if the generated errors by the computation, such as the round or the truncation ones, vanish when the computation advances in the mesh. A scheme is consistent if the local truncation errors obtainded when discretizing the Taylor series tend to zero when h, k and the elemental time interval tend to zero. The discretization error is a combination of the truncation error in the equation and the errors in the initial and boundary conditions.

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Last modified 4/2/2000 by Juan de Lara (Juan.Lara@ii.uam.es, http://www.ii.uam.es/~jlara) need help for using this courses?.

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