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The [[CPU]] time to solve the system of equations differs substantially from method to method. Finite differences are usually the cheapest on a per grid point basis followed by the finite element method and spectral method. However, a per grid point basis comparison is a little like comparing apple and oranges. Spectral methods deliver more accuracy on a per grid point basis than either [[FEM]] or [[FDM]]. The comparison is more meaningful if the question is recast as ”what is the computational cost to achieve a given error tolerance?”. The problem becomes one of defining the error measure which is a complicated task in general situations.
==Application of Finite Differences to [[ODE]]==
Let us consider the problem of solving the following [[PDE]]:
u<sub>t</sub> + cu<sub>x</sub> = νu<sub>xx</sub>, 0 ≤x ≤L
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:<math> u= \hat u_0 e ^{kt} </math>
==Forward Euler Approximation==
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