Midpoint

This subject has a long history and important early papers include those by Deijaguin and Landau [29] (see Ref. 30) and Langmuir [31]. As noted by Langmuir in 1938, the total force acting on the planes can be regarded as the sum of a contribution from osmotic pressure, since the ion concentrations differ from those in the bulk, and a force due to the electric field. The total force must be constant across the gap and since the field, d /jdx is zero at the midpoint, the total force is given the net osmotic pressure at this point. If the solution is dilute, then... 

This equation may be solved by the same methods as used with the nonreactive coupled-channel equations (discussed later in section A3.11.4.2). Flowever, because F(p, p) changes rapidly with p, it is desirable to periodically change the expansion basis set ip. To do this we divide the range of p to be integrated into sectors and within each sector choose a (usually the midpoint) to define local eigenfimctions. The coiipled-chaimel equations just given then apply withm each sector, but at sector boundaries we change basis sets. Let y and 2 be the associated with adjacent sectors. Then, at the sector boundary p we require... 

Figure 6, Equi-nonadiabatic coupling lines for the terms ri2(r,v) and r23(r,y) as calculated for the C2H molecule for a fixed C—C distauce, that is, rcc — 1-35 A. (a) Equi-non-adiabaric coupling term lines for the ri2(r,y). (b) Equi-non-adiabatic coupling term lines for X23(r,y). The Cartesian coordinates (x,y) are related to (q, 9) as follows x — qcosO] y —, sin9, where q and 9 are measured with respect to the midpoint between the two carbons.

The implicit-midpoint (IM) scheme differs from IE above in that it is symmetric and symplectic. It is also special in the sense that the transformation matrix for the model linear problem is unitary, partitioning kinetic and potential-energy components identically. Like IE, IM is also A-stable. IM is (herefore a more reasonable candidate for integration of conservative systems, and several researchers have explored such applications [58, 59, 60, 61]. 

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. 

Table 1. Stability Limits and Resonant Tiraesteps for the Ver-let and Implicit Midpoint Schemes...

To solve this system, we apply the implicit midpoint scheme (see system (10)) to system (24) and follow the same algebraic manipulation outlined in [71, 72] to produce a nonlinear system V45(y) = 0, where Y = (X + X )/2. This system can be solved by reformulating this solution as a minimization task for the dynamics function... 

M. Mandziuk and T. Schlick. Resonance in the dynamics of chemical systems simulated by the implicit-midpoint scheme. Chem. Phys. Lett., 237 525-535, 1995. 

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

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... 

The purpose of this paper is twofold (i) We summarize possible difficulties with the midpoint method (other than resonance instability, which has been treated extensively elsewhere) by looking at a simple (molecular) model problem, (ii) We investigate the suitability of some energy conserving methods. 

Time-reversible energy conserving methods can be obtained by appropriate modifications to the (time-reversible) midpoint method. Two such modifications are (i) scaling of the force field by a scalar such that total energy... 

Note that this latter method differs from the midpoint method, where one would use r(q +i/2) = (qn+i + qn)/2 instead of (7c) for r +i/2 in (7b). For highly oscillatory systems with k e, this can be a significant difference, because r is discretized directly in (7). An example in 4 below shows that the midpoint method can become unstable while (7) and (6) remain. stable. 

Here a symmetric projection step is used to enforce conservation of energy. Let a(g,p) and b q,p) be two vector-valued functions such that (p a q,p) + U q) b q,p)) is bounded away from zero. Then we propose the following modified midpoint method,... 

This latter modified midpoint method does work well, however, for the long time integration of Hamiltonian systems which are not highly oscillatory. Note that conservation of any other first integral can be enforced in a similar manner. To our knowledge, this method has not been considered in the literature before in the context of Hamiltonian systems, although it is standard among methods for incompressible Navier-Stokes (where its time-reversibility is not an issue, however). 

For highly oscillatory Hamiltonian systems, the best energy conserving midpoint variant that we are aware of is (6). In the sequel we therefore examine only its performance. 

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 interesting question is now what happens if the midpoint method is applied to the cartesian formulation (9) with Vi = V2 = 0. The equations are... 

To obtain the unconditional stability of the midpoint method in local coordinates, one would have to consider the decoupling transformation from cartesian to local coordinates for the discrete variables etc. But this transformation, which for the continuous variables is not constant, necessarily is in error which depends on k, not e. The stability properties of the discrete dynamical systems obtained by the midpoint discretization in the different sets of coordinatc.s may therefore be significantly different when it 3> e [3]. 

The gain in stability can now be interpreted as resulting from the direct midpoint discretization of the rapidly vibrating, local variable r, thus avoiding the potentially damaging discrete decoupling transformation. 

On the other hand, our computations using (15) indicate that the midpoint method becomes unstable for a > 1. 

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] ... 

U. Ascher and S. Reich. The midpoint scheme and variants for Hamiltonian systems advantages and pitfalls. SIAM J. Sci. Comput., to appear. 

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. 

The barriers in Fig, 4-10 are high because it is difficult to twist ethylene out of its normal planar conformation. The energy is the same at the midpoint and the end points in Fig, 4-10 because, on twisting an ethylene molecule 180" out of its normal conformation, one obtains a molecule that is indistinguishable from the original. The molecule has 2-foid torsional syinnteiry. 

Figure 4-10 [he Poteiitiiil Energy Form lor Ethylene. The midpoint of the range of (>) is (T and the end points -F180 . that is. [ a. Tt], The mid point and end points are identical by molecular symmetry. 

Figure 4-11 The Potential Energy Form for Ethane. The midpoint of the range of oj is m =0° and the end points are 180°. The end points and the minima are identical by molecular symmetry and correspond to the stable staggered form.

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