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Finite Differential Methods (FDM) are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.
The correlation coefficient, plus graphical methods to verify the validity of a linear relationship (which is what the correlation coefficient measures), and the appropriate tests of the statisitical significance of the correlation coefficient.
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There are both advantages and disadvantages of data collection methods in statistics. The main advantages are the metrics and correlation one can draw from statistics. The disadvantages stem from sampling errors.
Euler's Method (see related link) can diverge from the real solution if the step size is chosen badly, or for certain types of differential equations.