Explicit Euler

To check the effect of integration, the following algorithms were tried Euler, explicit Runge-Kutta, semi-implicit and implicit Runge-Kutta with stepwise adjustment. All gave essentially identical results. In most cases, equations do not get stiff before the onset of temperature runaway. Above that, results are not interesting since tubular reactors should not be... 

Second method consists of a straightforward discretization method first order (Euler) explicit in time and finite differences in space. Both the time step and the grid size are kept constant and satisfying the Courant Friedrichs Lewy (CFL) condition to ensure the stability of the calculations. To deal with the transport part we have considered the minmod slope limiting method based on the first order upwind flux and the higher order Richtmyer scheme (see, e.g. Quarteroni and Valli, 1994, Chapter 14). We call this method SlopeLimit. 

If a single-stage Euler explicit time-integration scheme is used, the updated moment set can be written as... 

Eq. (B.l). Thus, as a first step, we need to consider the volume-average form of Eq. (B.l) or, equivalently, the volume-average forms of the individual terms in Eqs. (B.2)-(B.5). Using a single-stage Euler explicit time-integration scheme (Leveque, 2002), the finite-volume expression for the updated NDF has the form ... 

P, Jy, and J , are the components of the total orbital angular momentum J of the nuclei in the IX frame. The Euler angles a%, b, cx appear only in the P, P and P angular momentum operators. Since the results of their operation on Wigner rotation functions are known, we do not need then explicit expressions in temis of the partial derivatives of those Euler angles. 

Runge-Kutta methods are explicit methods that use several function evaluations for each time step. Runge-Kutta methods are traditionally written for/(f, y). The first-order Runge-Kutta method is Euler s method. A second-order Runge-Kutta method is... 

While the general form of the generalized Euler s equation (equation 9.9) allows for dissipation (through the term Hifc) expression for the momentum flux density as yet contains no explicit terms describing dissipation. Viscous stress forces may be added to our system of equations by appending to a (momentarily unspecified) tensor [Pg.467]

While the matrix R does not explicitly involve the axis of rotation along which e extends, nor the angle of rotation 6, it is known (from a theorem of Euler s) that every R is equivalent to a single rotation definable geometrically by an e and a 0. 

Electron Nuclear Dynamics (48) departs from a variational form where the state vector is both explicitly and implicitly time-dependent. A coherent state formulation for electron and nuclear motion is given and the relevant parameters are determined as functions of time from the Euler equations that define the stationary point of the functional. Yngve and his group have currently implemented the method for a determinantal electronic wave function and products of wave packets for the nuclei in the limit of zero width, a "classical" limit. Results are coming forth protons on methane (49), diatoms in laser fields (50), protons on water (51), and charge transfer (52) between oxygen and protons. 

A method which represents a compromise between the explicit and implicit Euler... 

In the case of droplets and bubbles, particle size and number density may respond to variations in shear or energy dissipation rate. Such variations are abundantly present in turbulent-stirred vessels. In fact, the explicit role of the revolving impeller is to produce small bubbles or drops, while in substantial parts of the vessel bubble or drop size may increase again due to locally lower turbulence levels. Particle size distributions and their spatial variations are therefore commonplace and unavoidable in industrial mixing equipment. This seriously limits the applicability of common Euler-Euler models exploiting just a single value for particle size. A way out is to adopt a multifluid or multiphase approach in which various particle size classes are distinguished, with mutual transition paths due to particle break-up and coalescence. Such models will be discussed further on. 

For simple systems, the McDowell molecular-orbital technique would seem to be more time-consuming than that of SJG. In more complicated situations, however, this approach should lead to more accurate results, not only by using a Runge-Kutta rather than Euler method, but also by employing directly Ps(e), rather than its Fourier transform Gi(tX whose explicit form may not be known. 

Thus the implicit methods become slower and slower as the number of ODEs increases, despite the fact that large step sizes can be taken. Therefore plain old explicit Euler turns out to run faster than the impheit methods on many reahstically large problems, unless the stiffness of the system is very, very severe. We will talk more about this in Chap. 5. 

The explicit first-order Euler algorithm is used. The variables that we are solving for as functions of time are V and H. The right-hand sides of Eqs. (5.3) and (5.4) are the derivative functions. These are called VDOT and HDOT in the program. At the nth step in time... 

Note that the equations are given in explicit algebraic form. Comparing to problem (35), note that all additional constraints have disappeared and the differential equations have had a simple, low-order Euler discretization applied to them. However, as with (35) we note that the Lagrange function for this problem. 

The expression of Eq. (32) implies a rank one contraction of (L in parentheses in the superscript explicitly indicates that operator is given in the laboratory frame) with the rank two Wigner rotation matrices [Qml(0]> which describe the transformation from the molecule-fixed M, the principal frame of the dipole-dipole interaction) to the laboratory frame through the set of Euler angles ml- If the scalar term is neglected, then the formulations of Eqs. (31) and (32) are fully equivalent to that of Eq. (6). The expressions given by Eqs. (31) and (32) are convenient for the case of the electron spin treated as a part of the combined lattice. 

As indicated in Fig. 7, the next step after either an explicit or an implicit energy density functional orbit optimization procedure. For this purpose, one introduces the auxiliary functional Q[p(r) made up of the energy functional [p(r) 9 ]. plus the auxiliary conditions which must be imposed on the variational magnitudes. Notice that there are many ways of carrying out this variation, but that - in general - one obtains Euler-Lagrange equations by setting W[p(r) = 0. 

The critical nucleus, which can be found explicitly, is described by a homoclinic trajectory of the Euler-Lagrange equation ew = g w) (see, for instance Bates and Fife, 1993). The fact that this perturbation plays a role of a threshold is clear from Fig.9 which demonstrates extreme sensitivity of the problem to slight variations around the critical nucleus representing particular initial data (see Ngan and Truskinovsky (1996b) for details). 

Although not recommended for practical use, the classical Euler extrapolation is a convenient example to illustrate the basic ideas and problems of numerical methods. Given a point (tj y1) of the numerical solution and a step size h, the explicit Euler method is based on the approximation (yi+1 - /( i+l " 4 dy/dt to extrapolate the solution... 

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