Finite differencing
The first step in our numerical implementation of the solver is to discretize the spatial coordinates into grid points (or nodes). The one-dimensional case is shown in the figure below, where the x coordinate is discretized into Ngrid grid points:
If the distance between the grid points Δ x is constant, it can easily be seen that it is given by L/(Ngrid – 1). With this arrangement, the second order derivative of ϕ at x = x0 can then be approximated by an algebraic equation:
(6.9)
where i is then the grid point located at x0. This ...
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