Error Growth in Numerical Integration of Differential Equations

Let us emphasize that the issues arising in the design and analysis of numerical methods for molecular dynamics are slightly different than those confronted in other application areas. For one thing the systems involved are highly structured having conservation properties such as first integrals and Hamiltonian structure. We address the issues related to the inherent structure of the molecular N-body problem in both this and the next chapter wherein we shall learn that symplectic discretizations are typically the most appropriate methods. 

1 Error Growth in Numerical Integration of Differential Equations 

In this section we discuss the issues of convergence and accuracy in numerical integration methods for solving ordinary differential equations. 

In short, the error in the approximation obtained on [0, r] is reduced in direct proportion to the number of steps taken to cover this interval. Another way to say this is that e och, or, using the order notation, e = 0(h). Because the global error is of order h , where r= 1, we say that Euler s method is a first order method, or that it converges with order r = 1. 

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