Backward error analysis

E. Hairer and Ch. Lubich. The life-span of backward error analysis for numerical integrators. Numer. Math. 76 (1997) 441-462... 

S. Reich. Backward error analysis for numerical integrators. SIAM J. Numer. Anal., to appear, 1999. 

Backward Error Analysis for Hamiltonian Splitting Methods 103... 

Let us summarize the general situation with regard to backward error analysis. Assume a smooth differential equation system... 

The implication of backward error analysis is that the convergence order is still directly relevant in molecular simulations, even when the accuracy of trajectories (described by the convergence theorem) cannot be verified. We can think of the modified energy surface as a rippled version of the original. The order determining the allowed magnitude of the fiuctuation. 

In Collisional Verlet, the momenta are adjusted at the collision point to preserve the kinetic energy (and hence the total energy). It is possible to project to some other manifold using projection techniques like those mentioned previously, as discussed in [40]. In particular, one may use the backward error analysis to obtain a modified Hamiltonian Hh corresponding to the Verlet method with stepsize h, then to project during collisions not onto the energy surface, but onto the modified energy surface, so that... 

The perturbation term may be viewed as the finite truncation of the perturbative expansion obtained from the backward error analysis using the methods of the previous chapter, (r is the order of the numerical method.) Let us assume that the perturbed equation is a realistic model for the numerical solutions, then define the evolving density by... 

Now consider the application of the Symplectic Euler method, q + = q + hpn+i, Pn+1 = Pn - Mn- As wc know, the solution through any given initial point remains confined to an ellipse for all n defined by Hh(q,p) = Hh(qo,Po), where Hh is the perturbed Hamiltonian defined by the backward error analysis. This implies that a density which is a function of Hh will remain invariant under the associated Liouvilie equation. There are many solutions of Clp = 0, none of which is attractive. 

A benefit of respecting the symplectic structure lies in the prospect of using backward error analysis to analyze the behavior of the method. As shown in [40] it is possible to work out the terms of the perturbative expansion of the Nose-Poincare Hamiltonian, i.e. to calculate the expansion... 

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