Hyperbolicity

The hyperbolic cross section model can be generalized fiirther by introducing a fiinction/(A ) (AE = E - Eq) to describe the reaction cross section above a tln-eshold ... 

Ideally, the rods in a quadnipole mass filter should have a hyperbolic geometry, but more conunon is a set of four cylindrical rods separated by a distance 2r, where r I. I6r[((figure Bl.7.8). This anangement provides... 

Development of weighted residual finite element schemes that can yield stable solutions for hyperbolic partial differential equations has been the subject of a considerable amount of research. The most successful outcome of these attempts is the development of the streamline upwinding technique by Brooks and Hughes (1982). The basic concept in the streamline upwinding is to modify the weighting function in the Galerkin scheme as... 

In the earlier versions of the streamline upwinding scheme the modified weight function was only applied to the convection tenns (i.e. first-order derivatives in the hyperbolic equations) while all other terms were weighted in the usual manner. This is called selective or inconsistent upwinding. Selective upwinding can be interpreted as the introduction of an artificial diffusion in addition to the physical diffusion to the weighted residual statement of the differential equation. This improves the stability of the scheme but the accuracy of the solution declines. 

The first order derivative in Equation (2.80) corresponds to the convection in a field problem and the examples shown in Figure 2.26 illustraTes the ina bility of the standard Galerkin method to produce meaningful results for convection-dominated equations. As described in the previous section to resolve this difficulty, in the solution of hyperbolic (convection-dominated) equations, upwind-ing or Petrov-Galerkin methods are employed. To demonstrate the application of upwinding we consider the case where only the weight function applied to the first-order derivative in the weak variational statement of the problem, represented by Equation (2.82), is modified. 

The integrals in Equation (3.32) are found using a quadrature over the element domain The viscoelastic constitutive equations used in the described model are hyperbolic equations and to obtain numerically stable solutions the convection terms in Equation (3.32) are weighted using streamline upwinding as (inconsistent upwinding)... 

Differential methods - in these techniques the internal grid coordinates are found via the solution of appropriate elliptic, parabolic or hyperbolic partial differential equations. 

The most commonly used semiempirical for describing PES s is the diatomics-in-molecules (DIM) method. This method uses a Hamiltonian with parameters for describing atomic and diatomic fragments within a molecule. The functional form, which is covered in detail by Tully, allows it to be parameterized from either ah initio calculations or spectroscopic results. The parameters must be fitted carefully in order for the method to give a reasonable description of the entire PES. Most cases where DIM yielded completely unreasonable results can be attributed to a poor fitting of parameters. Other semiempirical methods for describing the PES, which are discussed in the reviews below, are LEPS, hyperbolic map functions, the method of Agmon and Levine, and the mole-cules-in-molecules (MIM) method. 

Systems of Logarithms. There are two common systems of logarithms in use (1) the natural (Napierian or hyperbolic) system which uses the base e = 2.71828. . . (2) the common (Briggsian) system which uses the base 10. 

The difference in exponentials which occurs in Eq. (2.21) is directly related to the hyperbolic sine function... 

Table 2.1 Some Useful Relationships Involving the Hyperbolic Sine and Inverse Hyperbolic Sine Function...

Introduction of the inverse hyperbolic sine function encourages us to take Eq. (2.24) a bit further and derive an expression for 17 itself. Before continuing, let us remember the following ... 

The description of phenomena in a continuous medium such as a gas or a fluid often leads to partial differential equations. In particular, phenomena of wave propagation are described by a class of partial differential equations called hyperbolic, and these are essentially different in their properties from other classes such as those that describe equilibrium ( elhptic ) or diffusion and heat transfer ( para-bohc ). Prototypes are ... 

Inverse Hyperbolic Functions If x = sinh y, then y is the inverse hyperbolic sine of x written y = sinh" x or arcsinh x. sinh" x = log x + + 1)... 

Hyperbolic The wave equation d u/dt = c d u/dx + d u/dy ) represents wave propagation of many varied types. 

An example of a linear hyperbolic equation is the adveclion equation for flow of contaminants when the x and y velocity components areii and i , respectively. 

Hyperbolic Equations The most common situation yielding hyperbohc equations involves unsteady phenomena with convection. Two typical equations are the convective diffusive equation... 

---