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The finite difference (FD) method is often used when a function $ f_j=f(x_j)$ sampled on a homogeneous mesh $ x_j=j\Delta x$ satisfies a differential equation that can simply be approximated with finite difference quotients (1.4.2#eq.2). In explicit schemes, the solution is calculated directly in terms of known quantities: explicit schemes are usually easy to implement in a program, but more delicate to execute. Implicit schemes are more robust at execution, but result in more complicated codes solving a linear system: this can be slow in 2 or higher dimensions and is then often advantageous to reconsider a more general finite elements solution on an inhomogeneous mesh.

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