Weinan E. Principles of Multiscale Modeling

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7.6. NOTES 347

Let F = (L

atom

− L

qqc

)(y

). Then

= 0, k ≤

= ε

−1

(G(2

) − G( 2D

)),

= ε

−1

(G(2

) + G(2D

−

) − 2G(2D

)),

= (L

atom

− L

)

, k ≥ 1.

(7.5.11)

Taylor expansion gives:











0, k ≤

O(1), k =

1, 0

O(ε

), k ≥ 1.

(7.5.12)

We see that the truncation error is small, except at the interface where it is O(1). Using

this fact, one can show under sharp stability conditions that the maximum error in the

deformation gradient is of O(ε). The details can be found in [33].

As we said earlier, what makes this kind of analysis possible is that the models used

in both the local and the nonlocal region can be regarded as consistent approximations to

the same continuum model. Extension of this kind of analysis to the case when statistical

errors are involved is still an open problem.

7.6 Notes

This is the most diﬃcult chapter to write, and perhaps many readers will not ﬁnd

the materials they expected to see. The diﬃculty is the following. On one hand, the

idea of developing hybrid formulations or hybrid algorithms that couple diﬀerent levels

of physical models together is extremely popular in many disciplines nowadays. On the

other hand, a lot of the existing work involves some ad hoc components in one way or

another. Except for very few cases, it is not clear how on e can assess the accuracy of

these hybrid schemes. Consequently, a description of these methodologies will inevitably

become a set of receipes rather than a coherent set of ideas with a solid foundation.

For this reason, we opt to discuss the general principles rather than the latest exciting

examples. Our selection of the gener al principles also reﬂect the desire to put things on

a solid foundation: After discussing the general coupling strategies, we chose to focus on

two classical issues in numerical analysis, namely, stability and accuracy.

348 CHAPTER 7. RESOLVING LOCAL EVENTS OR SINGULARITIES

The most unsatisfactory aspect of all this is the lack of understanding of the eﬀect

of noise or ﬂuctuations. We have seen from Ren’s work that the accumulation of noise

can lead to numerical instabilities. Bu t what about accuracy? There are examples which

demonstrate that the amplitude of the ﬂuctuations is reduced when kinetic Monte Carlo

models are replaced in part of the computational domain by deterministic continuum

equations [17, 43]. This suggests that instead of deterministic continuum models, one

should use stochastic continuum models when developing coupled atomistic-continuum

simulation methods [17, 43]. Doing this consistently is still a challenge.

Figure 7.6 is an image taken from [10] for the adiabatic piston problem. A long

quasi one-dimensional box with adiabatic walls is divided in two halves by a thermally-

insulating piston of mass M that can move along the length of the box without friction.

The two halves of the box is ﬁlled with ﬂuid particles of mass m (m ≪ M). Pressure

diﬀerence on both sides of the piston drives the piston to mechanical equilibrium, in

which the pressure diﬀerence vanishes. More interestingly, it has been understood th at

the mechanical asymmetric ﬂuctuations of the piston due to its thermal motion eventually

drives the system to thermal equilibrium, in which the temperature diﬀerence of the two

sides also vanishes. This process can in principle be simulated using molecular dynamics

or ﬂuctuating hydrodynamics. However, classical (deterministic) hydrodynamics fails to

predict the eventual full thermal equilibrium. The consequence of this is that the hybrid

method with deterministic hydrodynamics fails to predict the long term behavior of the

system accurately.

There is another diﬃculty. Take for example QM-MM methods at zero temperature.

In this case, we do not have to worry about noise. However, we still do not know how

to systematically quantify the error in QM-MM meth ods since we do not have a natural

small parameter to measure the size of the error. In classical numerical analysis, we

measure the numerical error in terms of the size of the grid. In homogenization theory,

we measure the error of the homogenized model in terms of the small parameter ε (see

Section 2.3). Here, th ere is no such a small parameter, not an obvious one at least.

7.6. NOTES 349

Figure 7.9: Relaxation of a massive rigid piston (M/m = 4000) which is initially not in

mechanical equilibrium. Shown in the ﬁgure is the position of the piston as a function

of time. Mechanical equilibrium is established through rapid oscillations. Both the

hybrid method with ﬂuctuating hydrodynamics and the direct particle simulation are

able to predict the very slow relaxation to the thermal equilibrium. The hybrid method

with deterministic hydrodynamics fails to do so or at least under-predicts the relaxation

rate. The inset highlights the initial oscillations and shows that in the regime where th e

ﬂuctuations do not matter the deterministic and stochastic hybrid methods agree well

(from [10], courtesy of John Bell.)

350 CHAPTER 7. RESOLVING LOCAL EVENTS OR SINGULARITIES

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