Errors discretization

Computational issues that are pertinent in MD simulations are time complexity of the force calculations and the accuracy of the particle trajectories including other necessary quantitative measures. These two issues overwhelm computational scientists in several ways. MD simulations are done for long time periods and since numerical integration techniques involve discretization errors and stability restrictions which when not put in check, may corrupt the numerical solutions in such a way that they do not have any meaning and therefore, no useful inferences can be drawn from them. Different strategies such as globally stable numerical integrators and multiple time steps implementations have been used in this respect (see [27, 31]). 

In most engineering problems the boundary of the problem domain includes curved sections. The discretization of domains with curved boundaries using meshes that consist of elements with straight sides inevitably involves some error. This type of discretization error can obviously be reduced by mesh refinements. However, in general, it cannot be entirely eliminated unless finite elements which themselves have curved sides are used. 

The discretization of a problem domain into a finite element mesh consisting of randomly sized triangular elements is shown in Figure 2,1. In the coarse mesh shown there are relatively large gaps between the actual domain boundary and the boundary of the mesh and hence the overall discretization error is expected to be large. 

Discretization error depends on the step size, i.e., if Ax. —> 0, the algorithm would theoretically be exact. The error for Euler method at step N is 0 N(Ax) and total accumulated error is 0 (Ax), that is, it is a first-order method. 

A direct computation of Eq (27) may reach accuracy up to the level of discrete error, but this needs multiplications plus (N-i) additions. For two-dimensional problem, it needs N XM multiplications and (W-1) X (M-1) additions. The computational work will be enormous for very large grid numbers, so a main concern is how to get the results within a reasonable CPU time. At present, MLMI and discrete convolution and FFT based method (DC-FFT) are two preferential candidates that can meet the demands for accuracy and efficiency. 

The first conclusion drawn from the comparison is that the numerical accuracy is dominated largely by the discrete errors. When the same influence coefficients were employed. 

The term numerical diffusion describes the effect of artificial diffusive fluxes which are induced by discretization errors. This effect becomes visible when the transport of quantities with small diffusivities [with the exact meaning of small yet to be specified in Eq. (42)] is considered. In macroscopic systems such small diffusivities are rarely found, at least when being looked at from a phenomenological point of view. The reason for the reduced importance of numerical diffusion in many macroscopic systems lies in the turbulent nature of most macro flows. The turbulent velocity fluctuations induce an effective diffusivity of comparatively large magnitude which includes transport effects due to turbulent eddies [1]. The effective diffusivity often dominates the numerical diffusivity. In contrast, micro flows are often laminar, and especially for liquid flows numerical diffusion can become the major effect limiting the accuracy of the model predictions. 

We replace the integral from Si to S2 by a sum over a regular grid. We do this by applying first a variable transformation (to be specified by some criteria) such that after this transformation an equidistant grid can be used. An estimate of the discretization error is possible by means of tricky and non-trivial application of analysis. Details on this are given in the appendix, which is a rather important part of this paper. 

The key feature is - both for the expansion of 1/r or e " in terms of even-tempered Gaussians - that, for large n, the cut-off error goes as exp(-anh) with h the step size and that the discretization errors goes as exp(—6/h), with a and b constants. While - for fixed n a small h is good for the discretization error, it is bad for the cut-off error and vice versa. The best compromise is that h 1/%/ni which implies that the overall error goes as exp(—Cy )-... 

Estimates for the discretization error are derived in the appendix. Unlike the estimates (2.6) these are not obtained as strict inequalities, but rather as leading terms of asymptotic expansions. For the integral (2.12a) with the integration limits —oo to oo the discretization error is (for large n and sufficiently small h, see appendix... 

The estimation of the discretization error is fortunately rather easy, relying on the results of appendix E (which contains the difficult part of the derivation). In fact the discretization error /a(r) given by (2.14) is simply proportional to 1/r. Hence... 

There is one difficulty insofar as (3.8) is only an estimate of the absolnte valne of the discretization error. It cannot be exclnded that (depending on how the limit xi —oo, X2 oo is performed, see appendix D) c and a have opposite sign. In this case the minimum absolute error may vanish, while (3109a) is still valid. 

The examples given in the appendix give some indications on the properties which the mapping function has to satisfy that both the cut-off error and the discretization error decrease exponentially (or faster) with nh and /h respectively and don t depend too strongly on r. Further studies are necessary to settle this problem. 

For quantum chemistry the expansion of e in a Gaussian basis is, of course, much more important than that of 1/r. The formalism is a little more lengthy than for 1/r, but the essential steps of the derivation are the same. For an even-tempered basis one has a cut-off error exp(—n/i) and a discretization error exp(-7//i), such that results of the type (2.15) and (2.16) result. Of course, e is not well represented for r very small and r very large. This is even more so for 1/r, but this wrong behaviour has practically no effect on the rate of convergence of a matrix representation of the Hamiltonian. This is very different for basis set of type (1.1). Details will be published elsewhere. 

We want to approximate the integral f x)dx by dividing the integration domain into n intervals of the same length h and by approximating /(x) in each interval by its value at the center of the interval. The discretization error is then... 

There is an alternative - and for our purposes more powerful - way to estimate the discretization error, namely in terms of the Fourier expansion of a periodic -function. We write hf xk), see (A.l), as [24]... 

We write ej, to indicate that this is a discretization error. 

We want to minimize e as function of h for fixed n. Since the discretization error only depends on h, it is obvious that one should make h as small as possible, in order to minimize it. We ean therefore assume that /z is so small that... 

Since Inn is a slowly varying function of n, the error goes essentially as n. This is the typical behaviour of a discretization error for a numerical integration [23], but is atypical for the examples that we want to study. 

While the functionin (B.l) is convex for all x, they( ) in (C.l) is concave from X = 0 to the inflection point x, = l/ /2 a and convex from Xi to oo. This means that the discretization error is negative for intervals between 0 and Xi and positive between Xj and oo, such that a partial cancellation of the error is possible. 

The discretization error Cd for finite integration limits yi and y2 contains in addition to (D.8) two extra terms (under the sum) that contain incomplete Gamma functions. We don t need their explicit form for the estimation of the dominating part of the overall error. Of course, expanding these extra terms in powers of h would lead to the error estimation (A.4), that holds for extremely small h (and sufficiently small /) which is rather irrelevant in the present context. 

Somewhat similar to appendix C we have a discretization error that goes as exp(—6//i) and a cut-off error exp(-anh). The minimum as function of h is achieved (for large n) if... 

This estimate of the discretization error ought to be known in numerical mathematics. Usually it is easier to derive formulas like this than too look them up in the literature. 

Equation (4.33) requires the computation of time derivatives. In molecular dynamics discretization errors due to the finite time step dt are of order Oidt2). Therefore we would like to estimate the time derivative in (4.33) with the same accuracy. 

Likewise, we can initialize the notional-particle properties so that both fxO, 0 and 6s (x, t) are null.145 The estimation error in the initial conditions, (6.222), is then due only to discretization error ... 

Separating I),p(x. t + At) into bias and discretization error then yields... 

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