Implicit midpoint method

A popular implicit discretization is the (implicit) midpoint method [7] which, applied to a system of the type... 

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] ... 

Jll 21 22 — fl22fll2. 10. Show that the implicit midpoint method... 

The Implicit Midpoint method which, for the ODE z = f(z) takes the form... 

This indieates that the Implicit Midpoint method has no stability threshold for the stepsize when solving a linear system This remarkable property may inspire some optimism that Implicit Midpoint would allow much larger stepsize in molecular dynamics simulation than does Verlet. 

Fig. 4.2 The frequency ratio 12/i2 for the Implicit Midpoint method. A frequency ratio of 1 indicates that the phase of the harmonic oscillator is accurate...

If the Implicit Midpoint method is viewed as the exact solution of a perturbed harmonic oscillator, then the frequency of this oscillator is... 

In the diagram below (Fig. 4.2), we plot the ratio against/t and S2. If this ratio is 1, the frequency is accurately represented. The figure shows that the frequency is only well approximated in case hQ is sufficiently small. For hQ 1 the frequency is highly modified. From this we might infer that a molecular dynamics simulation performed using the Implicit Midpoint method is only likely to be accurate in terms of resolution of temporally correlated quantities in case h 2 is relatively small. 

Implicit Midpoint Preserves First Integrals. Show that the implicit midpoint method exactly preserves quadratic first integrals. That is, if we have a matrix A ... 

The values q = 0,1/12,1/4 and 1/2 correspond, respectively [66], to the Ver-let, Stdrmer-Cowell/Numerov, implicit-midpoint, and LIM2 methods, the latter introduced in [41]. All integrators are second-order, except for Stbrmer-Cowell/Numerov, which is fourth-order accurate. 

Note that, in loeal eoordinates. Step 2 is equivalent to integrating the equations (13). Thus, Step 2 can either be performed in loeal or in eartesian coordinates. We consider two different implicit methods for this purpose, namely, the midpoint method and the energy conserving method (6) which, in this example, coineides with the method (7) (because the V term appearing in (6) and (7) for q = qi — q2 is quadratie here). These methods are applied to the formulation in cartesian and in local coordinates and the properties of the resulting propagation maps are discussed next. 

The form of the Hamiltonian impedes efficient symplectic discretization. While symplectic discretization of the general constrained Hamiltonian system is possible using, e.g., the methods of Jay [19], these methods will require the solution of a nontrivial nonlinear system of equations at each step which can be quite costly. An alternative approach is described in [10] ( impetus-striction ) which essentially converts the Lagrange multiplier for the constraint to a differential equation before solving the entire system with implicit midpoint this method also appears to be quite costly on a per-step basis. 

Various integration methods were tested on the dynamic model equations. They included an implicit iterative multistep method, an implicit Euler/modified Euler method, an implicit midpoint averaging method, and a modified divided difference form of the variable-order/variable-step Adams PECE formulas with local extrapolation. However, the best integrator for our system of equations turned out to be the variable-step fifth-order Runge-Kutta-Fehlberg method. This explicit method was used for all of the calculations presented here. 

When we apply a method such as Implicit Midpoint to a nonlinear molecular dynamics problem, we may reduce the equations to determining midpoint values q, p such that... 

If the parameters Aij, aij and Eij are all known, the initial concentrations and a temperature profile are given, the rate equations would predict the behaviour of the reaction. For very large systems a program LARKIN that integrates the, in general stiff, system of equations [27]. The initial value problems may be solved by routines like METANl [29] or SODEX [30, 31]. Both methods are based on a semi-implicit midpoint rule. 

In section 9.2 we illustrated one explicit method, Euler s forward method. In the present section, we likewise used only one type of implicit method, based on the trapezoidal or midpoint rule. All our examples have used constant increments Af higher computational efficiency can oftenbe obtained by making the step size dependent on the magnitudes of the changes in the dependent variables. Still, these examples illustrate that, upon comparing equivalent implicit and explicit methods, the former usually allow larger step sizes for a given accuracy, or yield more accurate results for the same step size. On the other hand, implicit methods typically require considerably more initial effort to implement. 

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