The Flow Map

We will introduce the following notation to describe the flow map of a Hamiltonian system with Hamiltonian H ... 

Typically, the expander has a separate flow and efficiency map as shown in Figure 8.11 although, a number of other options are available such as a torque map instead of efficiency (which may be better for use at low pressure ratios and speeds). For the expander, since the corrected flow is constant for most of the operating area (i.e. a zero slope of the flow map as shown in Figure 8.11), the best element representation is a flow element that is, obtain the flow based on pressure ratio and speed. 

Instead of polarized noble gases, thermally polarized NMR microimaging was used to study of liquid and gas flow in monolithic catalysts. Two-dimensional spatial maps of flow velocity distributions for acetylene, propane, and butane flowing along the transport channels of shaped monolithic alumina catalysts were obtained at 7 T by NMR, with true in-plane resolution of 400 xm and reasonable detection times. The flow maps reveal the highly nonuniform spatial distribution of shear rates within the monolith channels of square cross-section, the kind of information essential for evaluation and improvement of the efficiency of mass transfer in shaped catalysts. The water flow imaging, for comparison, demonstrates the transformation of a transient flow pattern observed closer to the inflow edge of a monolith into a fully developed one further downstream. 

In this paper experiments are described that expand the region of the pulsing flow mode from its naturally occurring window in the flow map to lower average gas and liquid flow rates Hanika et al. (1990) showed that such is possible by varying the liquid flow rate in time... 

Different flow regimes were described in early work by Weekman and Myers [22], who passed air and water downward through beds of glass beads or catalyst spheres. They presented the results as a flow map, an arithmetic plot of gas mass velocity versus liquid mass velocity, with lines marking regime boundaries. Results from a similar study by Tosun [23] are shown in Figure 8.11, where a log-log plot is used. Other workers have used liquid and gas superficial velocities or Reynolds numbers as the coordinates on the flow maps. 

The flow map for helium-methanol is similar to that for air-methanol but displaced to 10-fold-lower gas rates. This shows that the velocity of the gas is more important than the mass flow rate. The plot for helium would be close to that for air if based on linear velocity. In many high-pressure reactors, such as HDS or HDN reactors, the gas density is several times that of air at STP although some data are available, the effect of high gas density on the flow transitions is still uncertain [20,23]. 

The classical systems treated here can be solved forward or backward in time. Observe that = Id (the identity map), thus the flow map is invertible and,... 

The flow map provides an easy notation for describing solutions (even in cases where it is difficult or impossible to compute it exactly). When we speak of numerical methods for solving differential equations, we generally mean methods that approximate the flow map in a suitable sense. 

Despite the existence of a well-defined flow map, we cannot solve most systems of differential equations analytically (even Hamiltonian ones) so we cannot usually write out a formula for the flow map. An exception is the harmonic oscillator. 

Noting that the origin of time is arbitrary in an autonomous system such as this one, we may think of the exponential as a representation of the flow map,... 

In order to be able to easily compare the properties of different methods in a unified way, we focus in this chapter primarily on a particular class of schemes, generalized one-step methods. Suppose that the system under study has a well defined flow map , defined on the phase space (which is assumed to exclude any singular points of the potential energy function). The solution of the initial value problem, z = /(z), z(0) = may be written z(f,<) (with z(0,( ) = <), and the flow-map satisfies = z(f, 5). A one-step method, starting from a given point, approximates a point on the solution trajectory at a given time h units later. Such a method defines a map % of the phase space as illustrated in Fig. 2.1. 

Fig. 2.1 A step with the flow map approximation is illustrated in comparison to the corresponding step along the solution curve defined by the flow map...

The most straightforward rewriting of the Verlet method is to put the equations and the discretization in Hamiltonian form, i.e. introducing momenta/> = Mv, and thus pn = Mv , which results in the flow map approximation (taking us from any point in phase space (q,p) to a new point (Q, P) ... 

It is important in this definition that this is an equivalence that holds everywhere (or at least in some open set in W ), so the statement is not just that I z t)) = /(z(0)) for some particular trajectory, but, moreover, I is conserved for all nearby initial conditions. The flow map preserves the first integral, thus I(. i z)) = I(z). For example, in a Hamiltonian system, the flow map conserves the energy ... 

Consider a set of points f) in phase space with evolution associated to a differential equation z = /(z) described by the flow map f(S(0)) = > t). Liouville s theorem [ 16] states that the volume of such a set is invariant with respect to t if the divergence of / vanishes, i.e. 

Fig. 2.6 Filamentation of an evolving disk under the flow map of the Lennard-Jones osdllator. Lighter regions represent later images. Despite the increasing complexity of the shapes, each snapshot has the same area...

This is popularly referred to as the adjoint of although it seems something of an abuse of mathematical language to refer to it in this way. For the flow map we know that the inverse map is precisely. -h, so = h, i-e. the flow map is in the normal sense self-adjoint, i.e. symmetric. However, such a property does not hold in the general case. In particular, consider Euler s method... 

Symplectic integrators may be constructed in several ways. First, we may look within standard classes of methods such as the family of Runge-Kutta schemes to see if there are choices of coefficients which make the methods automatically conserve the symplectic 2-form. A second, more direct approach is based on splitting. The idea of splitting methods, often referred to in the literature as Lie-Trotter methods, is that we divide the Hamiltonian into parts, and determine the flow maps (or, in some cases, approximate flow maps) for the parts, then compose the maps to define numerical methods for the whole system. 

When using splittings, it is not necessary to solve each Hamiltonian of a splitting using the exact flow. Instead, we may replace the flow maps of any part by an approximation. More generally, if we have any two symplectic numerical methods, say and then the composition... 

This method requires that the positions (and forces) be known at two successive points h apart in time in order to initialize the iteration. These might be generated by using the Verlet method or some other self-starting scheme. Beeman s algorithm is explicit since, given q , q - andp , one directly obtains q + and then, q i, and thus p +i, with only one new force evaluation. Because it is a partitioned multistep method, its analysis is more involved than for the one-step methods, and, in particular its qualitative features are difficult to relate to those of the flow map. The order of accuracy of the scheme above can be shown to be three. 

Let/ satisfy the Lipshitz condition f u) -/(w) < L u - u. Let % represent the flow map approximation associated to Euler s method. Show that... 

In this chapter, we show that a symplectic integrator can be viewed as being effectively equivalent to the flow map of a certain Hamiltonian system. The starting point is that symplectic integrators are symplectic maps that are near to the identity since they depend on a parameter (the stepsize h) which can be chosen as small as needed, and, if consistent, in the limit -> 0, such a map must tend to the identity map. We can express the fundamental consequence as follows not only are Hamiltonian flow maps symplectic, but also near-identity symplectic maps are (in an approximate sense) Hamiltonian flow maps [31], The fact leads to the existence of a modifled (perturbed) Hamiltonian from which the discrete trajectory may be derived (as snapshots of continuous trajectories). In some cases we may derive this perturbed Hamiltonian as an expansion in powers of the stepsize. 

Now consider the more general Hamiltonian setting. Let 4 be a rth order symplectic integrator, r > 1. Suppose that it is the flow map of a certain Hamiltonian system, with Hamiltonian Hh. If the method order is r, we may expect this Hamiltonian to be a 0 W) approximation of H, thus we posit an expansion of the form... 

This gives a concise formula for the evolution of any function of the phase variables, including, in particular, any solution component z,-. Thus the flow map of the system can be represented by exp(t /). When one writes — exp(t /), what is actually meant is that the individual components satisfy... 

The flow map gives a solution of the differential equation, thus for a time-... 

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