Step size

Step size is critical in all simulations. This is the increment for integrating the equations of motion. It ultimately determines the accuracy of the numerical integration. For molecules with high frequency motion, such as bond vibrations that involve hydrogens, use a small step size. 

Temperature also determines step size. An acceptable time step for room temperature simulations is about 0.5-1 fs for All Atom systems or for simulations that do not constrain hydrogen atoms. For United Atom systems or systems containing only heavy atoms, you can use steps of 1-2 fs. 

The temperature of a simulation depends on your objectives. You might use high temperatures to search for additional conformations of a molecule (see Quenched Dynamics on page 78). Room temperature simulations generally provide dynamic properties of molecules such as proteins, peptides, and small drug molecules. Low temperatures ( 250 K) often promote a molecule to a lower energy conformation than you could obtain by geometry optimization alone. 

One of the features of the Transient Analysis that causes much confusion is the Maximum step size argument in the Transient dialog box. Suppose we wish to run a circuit that has a sinusoidal source. We expect the voltages and currents to look sinusoidal, as follows  

When PSpice runs a Transient Analysis, it solves differential equations to find voltages and currents versus time. The time between simulation points is chosen to be as large as possible while keeping the simulation error below a specified maximum. In some cases, where PSpice can take large time steps, you may get a graph that does not look sinusoidal  

The graph does not look much like a sine wave because the time between points is so large. If we decrease the time between points, we see that the points do lie on a sinusoidal curve. The graph below is an overlay plot of the detailed sine wave of the top graph (the trace had a very small time step) and the center graph showing a sine wave trace with a large time step  

Monti Ota/Wortt Com IPwwiweic Sweep ITTemperaaira (Swoop) ISave Bias Point ILood Bias Point 

The Maximum Step Size is initially blank so that PSpice will choose as large a time step as possible. If you need more points in your simulation, you must enter a number for the Maximum Step Size. 

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This value determines the amount the step-size is reduced to satisfy the criteria of a SSQ which decreases from one iteration to the next. The amount of the decrease is equal to the previous value of the step-limiting parameter divided by RP. 

Simulations of the adaptive reconstruction have been performed for a single slice of a porosity in ferritic weld as shown in Fig. 2a [11]. The image matrix has the dimensions 230x120 pixels. The number of beams in each projection is M=131. The total number of projections K was chosen to be 50. For the projections the usual CT setup was used restricted to angels between 0° and 180° with the uniform step size of about 3.7°. The diagonal form of the quadratic criteria F(a,a) and f(a,a) were used for the reconstruction algorithms (5) and (6). 

The measurements were made along the cracks with an average step size of 3 mm. The predictions were calculated from a position -15 mm to + 15 ram for set 1, from -40 mm to + 40 mm for set 2 and from -25 mm to + 25 mm for set 3. The impedance change has been calculated at 1mm intervals in the range. Taking into account the symmetry of the configuration, only half of the predictions need to be calculated. 

A8, which leads to D, = 1/(2A8). The factor of two arises because a minimum of two data points per period are needed to sample a sinusoidal wavefonn. Naturally, the broadband light source will detennine the actual content of the spectrum, but it is important that the step size be small enough to acconunodate the highest frequency components of the source, otherwise they... 

If the computed step size exceeds the trust radius, (, its direction is reoptunized under the condition that Aq = t, i.e., the Lagrangian... 

E. Hairer and D. Stoffer. Reversible long-term integration with variable step sizes. Report (1995)... 

It is also interesting to examine the behavior of the optimized solution as a function of the step size. As discussed below, the proposed algorithm is very... 

For example, the SHAKE algorithm [17] freezes out particular motions, such as bond stretching, using holonomic constraints. One of the differences between SHAKE and the present approach is that in SHAKE we have to know in advance the identity of the fast modes. No such restriction is imposed in the present investigation. Another related algorithm is the Backward Euler approach [18], in which a Langevin equation is solved and the slow modes are constantly cooled down. However, the Backward Euler scheme employs an initial value solver of the differential equation and therefore the increase in step size is limited. 

In this paper, we discuss semi-implicit/implicit integration methods for highly oscillatory Hamiltonian systems. Such systems arise, for example, in molecular dynamics [1] and in the finite dimensional truncation of Hamiltonian partial differential equations. Classical discretization methods, such as the Verlet method [19], require step-sizes k smaller than the period e of the fast oscillations. Then these methods find pointwise accurate approximate solutions. But the time-step restriction implies an enormous computational burden. Furthermore, in many cases the high-frequency responses are of little or no interest. Consequently, various researchers have considered the use of scini-implicit/implicit methods, e.g. [6, 11, 9, 16, 18, 12, 13, 8, 17, 3]. 

W, g potential functions, k 1, has been discussed in various papers (see, for example, [6, 11, 9, 16, 3]). It has been pointed out that, for step-sizes /j > e = 1/ /k, the midpoint method can become unstable due to resonances [9, 16], i.e., for specific values of k. However, generic instabilities arise if the step-size k is chosen such that is not small [3, 6, 18], For systems with a rotational symmetry this has been shown rigorously in [6j. This effect is generic for highly oscillatory Hamiltonian systems, as argued for in [3] in terms of decoupling transformations and proved for a linear time varying system without symmetry. 

In general, the solution components of the DAE (4) are the correct limits (as K —> oo) of the corresponding slowly varying solution components of the free dynamics only if an additional (conservative) force term is introduced in the constrained system [14, 5]. It turns out [3] that the midpoint method may falsely approximate this correcting force term to zero unless k — 0 e), which leads to a step-size restriction of the same order of magnitude as explicit... 

Another option is a q,p) = p and b q,p) = VU q). This guarantees that we are discretizing a pure index-2 DAE for which A is well-defined. But for this choice we observed severe difficulties with Newton s method, where a step-size smaller even than what is required by explicit methods is needed to obtain convergence. In fact, it can be shown that when the linear harmonic oscillator is cast into such a projected DAE, the linearized problem can easily become unstable for k > . Another way is to check the conditions of the Newton-Kantorovich Theorem, which guarantees convergence of the Newton method. These conditions are also found to be satisfied only for a very small step size k, if is small. 

The standard discretization for the equations (9) in molecular dynamics is the (explicit) Verlet method. Stability considerations imply that the Verlet method must be applied with a step-size restriction k < e = j2jK,. Various methods have been suggested to avoid this step-size barrier. The most popular is to replace the stiff spring by a holonomic constraint, as in (4). For our first model problem, this leads to the equations d... 

Another way to overcome the step-size restriction fc < is to use multiple-time-stepping methods [4] or implicit methods [17, 18, 12, 3). In this paper, we examine the latter possibility. But for large molecular systems, fully implieit methods are very expensive. For that reason, we foeus on the general class of scmi-implicit methods depicted in Fig. 1 [12]. In this scheme. Step 3 of the nth time step ean be combined with Step 1 of the (n - - l)st time step. This then is a staggered two-step splitting method. We refer to [12] for further justification. 

As remarked earlier, we are interested in the behavior of this approximation for step-sizes k much larger than the period of the fast bond vibrations,... 

Unfortunately, discretization methods with large step sizes applied to such problems tend to miss this additional force term [3]. Furthermore, even if the implicit midpoint method is applied to a formulation in local coordinates, similar problems occur [3]. Since the midpoint scheme and its variants (6) and (7) are basically identical in local coordinates, the same problem can be expected for the energy conserving method (6). To demonstrate this, let us consider the following modified model problem [13] ... 

To be more precise, this error occurs in the limit /c — oo with Ef = 0(1) and step-size k such that k /ii = const. 3> 1. This error does not occur if Ef = 0 for the analytic problem, i.e., in case there is no vibrational energy in the stiff spring which implies V,. = U. 

Note that there are also variations in total energy which might be due to the so called step size resonance [26, 27]. Shown are also results for fourth order algorithm which gives qualitatively the same results as the second order SISM. This show that the step size resonances are not due to the low order integration method but rather to the symplectic methods [28]. 

Abstract. The overall Hamiltonian structure of the Quantum-Classical Molecular Dynamics model makes - analogously to classical molecular dynamics - symplectic integration schemes the methods of choice for long-term simulations. This has already been demonstrated by the symplectic PICKABACK method [19]. However, this method requires a relatively small step-size due to the high-frequency quantum modes. Therefore, following related ideas from classical molecular dynamics, we investigate symplectic multiple-time-stepping methods and indicate various possibilities to overcome the step-size limitation of PICKABACK. 

Here we suggest a different approach that propagates the system using multiple step-sizes, i.e., few steps with step-size At are taken in the slow classical part whereas many smaller steps with step-size 5t are taken in the highly oscillatory quantum subsystem (see, for example, [19, 4] for symplectic multiple-time-stepping methods in the context of classical molecular dynamics). Therefore, we consider a splitting of the Hamiltonian H = Hi +H2 in the following way ... 

For stability reasons, the micro-step-size 5t has to be chosen smaller than the inverse of the largest eigenvalue of the (scaled) truncated quantum operator % This can imply a very small value of 5t compared to... 

R. D. Skeel and J. J. Biesiadecki, Symplectic integrations with variable step-size , Annals Numer. Math., 191-198, 1994. 

High tern perature sim ulation s requ ire special con sideratiori ir choosing the sampling interval (see Step size" on page 89),... 

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