Flow map approximation

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) ... 

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

Figure 5 Map showing the Norman Landfill (USA) site. Transect A-A runs from MLS 35 to MLS 80 in the direction of groundwater flow. The approximate extent of the plume was determined from geophysical measurements...

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. 

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. 

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... 

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 bound tends to zero extremely rapidly (more rapidly than any power of h) as h 0. The assumption is that if this error in the approximation of the numerical map by the flow of a truncated perturbed Hamiltonian flow map is much smaller than the other errors present in our model, any drift due to lack of convergence would be minor, perhaps even invisible, on the timescale of simulation. 

To make this practical in the general case, where the Hamiltonian system is not integrable (and we cannot explicitly compute its flow map), we need to be able to compute (or approximate) the maps involved. 

Usually) time stepsize used for a numerieal method Approximate flow map defined by a numerieal method Divergenee of veetor field/... 

This section deals with the question of how to approximate the essential features of the flow for given energy E. Recall that the flow conserves energy, i.e., it maps the energy surface Pq E) = x e P H x) = E onto itself. In the language of statistical physics, we want to approximate the microcanonical ensemble. However, even for a symplectic discretization, the discrete flow / = (i/i ) does not conserve energy exactly, but only on... 

Approximate prediction of flow pattern may be quickly done using flow pattern maps, an example of which is shown in Fig. 6-2.5 (Baker, Oil Gas]., 53[12], 185-190, 192-195 [1954]). The Baker chart remains widely used however, for critical calculations the mechanistic model methods referenced previously are generally preferred for their greater accuracy, especially for large pipe diameters and fluids with ysical properties different from air/water at atmospheric pressure. In the chart. 

An expander performanee map of shaft power versus mass flow with lines of eonstant inlet temperature and pressure shall be provided. There shall be a minimum of four eonstant pressure lines with inerements of approximately 5 psi and a minimum of four eonstant temperature lines with inerements of approximately 100°F. The map shall be valid for rated speed with normal exhaust pressure and gas eomposition. The normal operating point shall be indieated on the map. 

The regions over which the different types of flow can occur are conveniently shown on a Flow Pattern Map in which a function of the gas flowrate is plotted against a function of the liquid flowrate and boundary lines are drawn to delineate the various regions. It should be home in mind that the distinction between any two flow patterns is not clear-cut and that these divisions are only approximate as each flow regime tends to merge in with its neighbours in any case, the whole classification is based on highly subjective observations. Several workers have produced their own maps 4 8. ... 

Distributions of water and reactants are of high interest for PEFCs as the membrane conductivity is strongly dependent on water content. The information of water distribution is instrumental for designing innovative water management schemes in a PEFC. A few authors have studied overall water balance by collection of the fuel cell effluent and condensation of the gas-phase water vapor. However, determination of the in situ distribution of water vapor is desirable at various locations within the anode and cathode gas channel flow paths. Mench et al. pioneered the use of a gas chromatograph for water distribution measurements. The technique can be used to directly map water distribution in the anode and cathode of an operating fuel cell with a time resolution of approximately 2 min and a spatial resolution limited only by the proximity of sample extraction ports located in gas channels. 

The most reliable methods for fully developed gas/liquid flows use mechanistic models to predict flow pattern, and use different pressure drop and void fraction estimation procedures for each flow pattern. Such methods are too lengthy to include here, and are well suited to incorporation into computer programs commercial codes for gas/liquid pipeline flows are available. Some key references for mechanistic methods for flow pattern transitions and flow regime-specific pressure drop and void fraction methods include Taitel and Dukler (AIChEJ., 22,47-55 [1976]), Barnea, et al. (Int. J. Multiphase Flow, 6, 217-225 [1980]), Barnea (Int. J. Multiphase Flow, 12, 733-744 [1986]), Taitel, Barnea, and Dukler (AIChE J., 26, 345-354 [1980]), Wallis (One-dimensional Two-phase Flow, McGraw-Hill, New York, 1969), and Dukler and Hubbard (Ind. Eng. Chem. Fun-dam., 14, 337-347 [1975]). For preliminary or approximate calculations, flow pattern maps and flow regime-independent empirical correlations, are simpler and faster to use. Such methods for horizontal and vertical flows are provided in the following. 

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