Characteristics of Methods

For a simple wave, apphcation of the method of characteristics (hodograph transformation) gives... 

Aris and Amundson (1958) solved the coupled, time-dependent material and energy balances, linearizing the equations about the operating point by a Taylor series expansion. This made the solution possible by the method of characteristic equations. The solution yielded two equations, one the slope condition and the other recognized by Gilles and Hofmann (1961) as the condition that sets the limits to avoid rate oscillation. This is called the... 

The most effective techniques for hyperbolic partial differential equations are based on the method of characteristics [19] and an extensive treatment of this method may be found in the literature of compressible fluid flow and plasticity fields. 

Abbott, M. B., An Introduction to the Method of Characteristics, American Elsevier Publishing, 1966. 

Population Density Response Surface. The algorithm method of characteristic is used to reduce the partial differential Equation (1) into a set pf coupled ordinary differential equations. Since T(n,t) is an exact differential, then... 

In order to solve Eq. (34), we use the method of characteristics and consider a family of classical trajectories on the inverted potential q(p, x), p(P, x), where P is an (A — 1)-dimensional parameter to characterize the trajectory and x is the time running for the infinite interval along the trajectory, where x = — oo corresponds to the minimum of the potential q(p, —oo) = q ,p(p, —oo) = 0. The solution we want is the trajectory that connects the two potential minima and along which the action becomes minimum. This is called the instanton trajectory and belongs to the above mentioned family qo(x) = q(Po At q close to the potential minimum q , the momentum p(q) is linear with respect to the deviation (q — q ) and Wo(q) is quadratic. 

In terms of transient behaviors, the most important parameters are the fluid compressibility and the viscous losses. In most field problems the inertia term is small compared with other terms in Eq. (128), and it is sometimes omitted in the analysis of natural gas transient flows (G4). Equations (123) and (128) constitute a pair of partial differential equations with p and W as dependent variables and t and x as independent variables. The equations are hyperbolic as shown, but become parabolic if the inertia term is omitted from Eq. (128). As we shall see later, the hyperbolic form must be retained if the method of characteristics (Section V,B,1) is to be used in the solution. 

Probably the least flexible of all methods with respect to the time-step and distance relationship is the method of characteristics (MOC). It requires the pipe lengths in a network to be adjusted to satisfy the condition of a common time interval, but provides an accurate solution of the differential equations. MOC has been successfully implemented by Goacher (G4), Streeter and associates (S6), and Masliyah and Shook (M5). More recently,... 

In contrast to the method of characteristics, which gives faithful simulation of transient flows but which is very restrictive in time step sizes, the stability of the implicit methods permit large time steps and drastic reduc-... 

While nonlinear in g2 (the coefficients 7,.r and b are lengthy integral expressions), the partial differential equation is linear in the derivatives. It can thus be solved by the method of characteristics, with the boundary conditions given by the coupling at /x = 0, as obtained from finite-T lattice QCD. 

By the method of characteristics 5, equation 17.121 may be reduced to an ordinary differential equation to give ... 

ACRIVOS, A. Ind. Eng. Chem. 45 (1956) 703. Method of characteristics technique. Application to heat and mass transfer problems. 

The simulation of a continuous, evaporative, crystallizer is described. Four methods to solve the nonlinear partial differential equation which describes the population dynamics, are compared with respect to their applicability, accuracy, efficiency and robustness. The method of lines transforms the partial differential equation into a set of ordinary differential equations. The Lax-Wendroff technique uses a finite difference approximation, to estimate both the derivative with respect to time and size. The remaining two are based on the method of characteristics. It can be concluded that the method of characteristics with a fixed time grid, the Lax-Wendroff technique and the transformation method, give satisfactory results in most of the applications. However, each of the methods has its o%m particular draw-back. The relevance of the major problems encountered are dicussed and it is concluded that the best method to be used depends very much on the application. 

Mihailescu-Suliciu, M., and Suliciu, I., 1992, On the method of characteristics in rate type viscoelasticity with non-monotone equilibrium curve, ZAMM 72 667. 

In the first approach we use the package FreeFem++ by Pironneau, Hecht and Le Hyaric. For more information we refer to http //www.freefem.org/ff++/. For the problem (6)-(10) the method of characteristics from Pironneau (1988) is used. We present a very short description of the method ... 

The first order operator is discretized using the method of characteristics. More precisely, the Equation (6) is written as ... 

Let c be an approximation for the solution c at a time m5f. Then the one step backward convection scheme by the method of characteristics reads as follows ... 

Pergamon Press, NY(1962), Chapter 26 "Solution of Characteristics of the Equations of One Dimensional Unsteady Flow , by A.E. Glennie (Method of Characteristics is described) Chapter 27 "Finite Difference Methods for One-Dimensional Unsteady Flow , by N.E. Hoskin B.W. Pearson (Mesh Method is described) 9) H.D. Huskey G,A. Korn, "Computer Handbook , McGraw-Hill, NY(1962)... 

Ibid, Vol 3(1964), "Fundamental Methods in Hydrodynamics , which includes "Hydro-dynamic Code HEMP , by M.L. Wilkins, pp 211-63 and "Method of Characteristics , by N.E. Hoskin, pp 265-93 15a) V. Lindner,... 

Systems , Ibid, p 538-52 [Use of the "Method of Characteristics in the code NIP (Normal Initiation Program) is described and compared with the "Mesh Methods . Method of Characteristics was described by Courant Friedrichs (Ref 6) and Glennie (Ref 8). 

The value 3 can be used for n in obtaining a solution of hydrodynamic equations for an ideal gas by the method of characteristics since this value makes for easy solution. These solutions describe roughly the propagation of finite waves thru a solid since most solids follow approximately the ideal gas adiabat with y = 3. This value of y is not to be confused with the true ratio of specific heats, which lies between 1 and 1-2/3 (Ref 10, p 184 Ref 8e, p 740)... 

The region of shock formation, represented in Fig for simplicity by a single point S, can then be constructed by the "method of characteristics described in the book of Courant Friedrichs (Ref 3). The pertinent velocities are given as equations 12,... 

Before comparing the above theoretical results with experiment, let us return briefly to the question of non-rigidity of the medium in contact with an exploding HE slab. Erkman (Ref 6) using a numerical procedure based on the method of characteristics obtained values of Ic for various media in contact with EL506D sheet explosive containing about 70% PETN 30% binder whose p0 = 1.4g/cc D = 7100 m/sec... 

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