Molecular dynamics numerical integration errors

In the previous chapter, we discussed the growth of error in numerical methods for differential equations. We saw that if the time interval is fixed, the error obeys the power law relationship with stepsize that is predicted by the convergence theory. We also saw that this did not contradict the exponential growth in the error with time (when the stepsize is fixed). The latter issue casts doubt on the reUance on the convergence order as a means for assessing the suitability of an integrator for molecular dynamics. 

It is possible to derive numerical methods that exactly (i.e. to within rounding error) conserve the energy at each step and this may seem a more natural approach to molecular dynamics than symplectic integration since we have discussed and evaluated methods in terms of their energy conservation. 

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