Discrete methods

In this paper, we discuss semi-implicit/implicit integration methods for highly oscillatory Hamiltonian systems. Such systems arise, for example, in molecular dynamics [1] and in the finite dimensional truncation of Hamiltonian partial differential equations. Classical discretization methods, such as the Verlet method [19], require step-sizes k smaller than the period e of the fast oscillations. Then these methods find pointwise accurate approximate solutions. But the time-step restriction implies an enormous computational burden. Furthermore, in many cases the high-frequency responses are of little or no interest. Consequently, various researchers have considered the use of scini-implicit/implicit methods, e.g. [6, 11, 9, 16, 18, 12, 13, 8, 17, 3]. 

Unfortunately, discretization methods with large step sizes applied to such problems tend to miss this additional force term [3]. Furthermore, even if the implicit midpoint method is applied to a formulation in local coordinates, similar problems occur [3]. Since the midpoint scheme and its variants (6) and (7) are basically identical in local coordinates, the same problem can be expected for the energy conserving method (6). To demonstrate this, let us consider the following modified model problem [13] ... 

The choice of parameterization and the design of a discretization method are not independent Some choices of parameters will facilitate symplec-tic/reversible discretization while others may make this task very difficult or render the resulting scheme practically useless because of the computational expense involved. 

Discretization of the governing equations. In this step, the exact partial differential equations to be solved are replaced by approximate algebraic equations written in terms of the nodal values of the dependent variables. Among the numerous discretization methods, finite difference, finite volume, and finite element methods are the most common. Tlxe finite difference method estimates spatial derivatives in terms of the nodal values and spacing between nodes. The governing equations are then written in terms of... 

In principle, this method is more accurate than the direct derivative discretization method described in the previous subsection, because the derivative calculations are made several times for the same surface point. Nevertheless, the method suffers essentially the same problems related to the finite derivative discretization on the lattice as in the previous case. 

Commercially available CFD codes use one of the three basic spatial discretization methods finite differences (FD), finite volumes (FV), or finite elements (FE). Earlier CFD codes used FD or FV methods and have been used in stress and flow problems. The major disadvantage of the FD method is that it is limited to structured grids, which are hard to apply to complex geometries and... 

V, ip, x, and t) in the PDF transport equation makes it intractable to solve using standard discretization methods. Instead, Lagrangian PDF methods (Pope 1994a) can be used to express the problem in terms of stochastic differential equations for so-called notional particles. In Chapter 7, we will discuss grid-based Eulerian PDF codes which also use notional particles. However, in the Eulerian context, a notional particle serves only as a discrete representation of the Eulerian PDF and not as a model for a Lagrangian fluid particle. The Lagrangian Monte-Carlo simulation methods discussed in Chapter 7 are based on Lagrangian PDF methods. 

The problem can be solved using the discretization method discussed in chapter 4 of this book. First candidates are generated using the normal boiling points and the normal boiling point. In the next stage, the remaining properties are calculated only for the molecules that have not been discarded. 

The discrete method has the advantage that samples can he processed at a high rate. For example, commercial colorimetric analysers are capable of yielding between 100 and 300 measurements per hour, whereas for continuous analysers a processing rate of 20— 80 samples per hour is normal. However, the high-throughput discrete analysers are appreciably more expensive than the continuous analysers. 

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. 

This discretization method obeys a conservation property, and therefore is called conservative. With the exception of the first element and the last element, every element face is a part of two elements. The areas of the coincident faces and the forces on them are computed in exactly the same way (except possibly for sign). Note that the sign conventions for the directions of the positive stresses is important in this regard. The force on the left face of some element is equal and opposite to the force on the right face of its leftward neighbor. Therefore, when the net forces are summed across all the elements, there is exact cancellation except for the first and last elements. For this reason no spurious forces can enter the system through the numerical discretization itself. The net force on the system of elements must be the net force caused by the boundary conditions on the left face of the first element and the right face of the last element. 

Unfortunately, resolution of the conservation problem requires knowledge of species flux, and hence details of the specific problem and discretization method. Therefore it is not possible in the general setting of the present discussion to give a universal solution. Nevertheless, a software author and users of a simulation code must be aware of the difficulty, and consider its resolution when setting up the difference approximations to the particular system of conservation equations. 

To be successful in solving applied and mostly differential problems numerically, we must know how to implement our physico-chemical based differential equations models inside standard numerical ODE solvers. The numerical ODE solvers that we use in this book are integrators that work only for first-order differential equations and first-order systems of differential equations. [Other DE solvers, for which we have no need in this book, are discretization methods, finite element methods, multigrid methods etc.]... 

When Bernie Shizgal arrived at UBC in 1970, his research interests were in applications of kinetic theory to nonequilibrium effects in reactive systems. He subsequently applied kinetic theory methods to the study of electron relaxation in atomic and molecular moderators,46 hot atom chemistry, nucleation,47 rarefied gas dynamics,48 gaseous electronics, and other physical systems. An important area of research has been the kinetic theory description of the high altitude portion of planetary atmospheres, and the escape of atmospheric species.49 An outgrowth of these kinetic theory applications was the development of a spectral method for the solution of differential and integral equations referred to as the quadrature discretization method (QDM), which has been used with considerable success in statistical, quantum, and fluid dynamics.50... 

B. D. Shizgal and H. Chen,/. Chem. Phys., 107,8051 (1997). The Quadrature Discretization Method (QDM) in the Solution of the Fokker-Planck Equation with Nonclassical Basis Functions. 

H.A. Mang and F.G. Rammerstorfer (eds.) IUTAM Symposium on Discretization Methods in Structural Mechanics. Proceedings of the IUTAM Symposium held in Vienna, Austria. 1999... 

In contrast with discrete methods, the thermal average is introduced in the continuum approach at the beginning of the procedure. Computer information on the distribution functions and related properties could be used (and in some cases are actually used), but in the standard formulation the input data only include macroscopic experimental bulk properties, supplemented by geometric molecular information. The physics of the system permits the use of this approximation. In fact the bulk properties of the solvent are slightly perturbed by the inclusion of one solute molecule. The deviations from the bulk properties (which become more important as the mole ratio increases) are small and can be considered at a further stage of the development of the model. 

The usual discretization methods for integral equations (collocation vs Galerkin, boundary elements) are presented in Section 1.2.5. 

In this example, one periodic element (a cross-over) of the laboratory scale version of Katapak -S was selected for the detailed CFD simulation with CFX-5. This solver uses the finite volume discretization method in combination with hybrid unstructured grids. Around 1,100 spherical particles of 1 mm diameter were included in the computational domain. As the liquid flows through the catalyst-filled channels at operating conditions below the load point (cf. Moritz and Hasse, 1999), permeability of the channel walls made of the wire mesh is not taken into account by this particular model. The catalyst-filled channels are considered fully wetted by the liquid creeping down, whereas the empty channels are completely occupied by the counter-current gas. It means that the bypass flow... 

Menzies, M.A. and Johnson, A.I., "Synthesis of Optimal Energy Recovery Networks Using Discrete Methods," Can. J. Chem. Eng. 50, 290, 1972. 

We now return to the loop-after-loop SESE calculations in [11]. The first two terms of the potential expansion Eq. (2), ZP and OP terms were evaluated in momentum space. For this purpose the Fourier transform was performed for the bound state wave functions n) in coordinate space. The latter were evaluated by the space discretization method. The MP term was calculated entirely in coordinate space. 

The numerical solution for the solute-humic cotransport model was obtained by an unconditionally stable, fully implicit finite difference discretization method. The three governing transport Eqs. (38), (48), and (54) in conjunction with the initial and boundary conditions given by Eqs. (39)-(41), (51)—(53), (58) and (59) were solved simultaneously [57]. All flux boundary conditions were estimated using a second-order accurate one sided approximation [53]. 

In the present paper the Boundary Finite Element Method is presented as a boundary discretization method for the numerical investigation of interfacial stress concentrations in composite laminates. In contrast to the classical boundary element method, the element formulation is finite element based, which avoids the necessity of a fundamental solution. Comparative results from finite element calculations show good agreement both for the laminate free-edge effect and for the example of the stress concentrations near cracks in composite laminates. 

For numerical investigations of stress localizations in laminates, the discretizational effort can be reduced significantly if only the boundary needs to be discretized, as it is for e -ample the case in the classical boundary element method (BEM). But in this method a fundamental solution is needed which is in many cases difficult to achieve or even unknown. The Boundary Finite Element Method (BFEM) to be presented here does not require such a fundamental solution, because the element formulation is based on the finite element method (FEM), Thus the BFEM can be characterized to be a finite element based boundary discretization method. This method was originally developed from Wolf and Song [10] under the name Consistent Finite Element Cell Method for time-dependent problems in soil-mechanics. The basic assumption of this method is that a stiffness matrix describing the force-displacement relation at discrete degrees of freedom at the boundary of the continuum is scalable with respect to one point in three-dimensional space, the so-called similarity center, if similar contours within the continuum are considered. In contrast to this, the current work deals with the case of equivalent cross-sectional properties, i.e., that cross-sections parallel to the boundary can be described by the same stiffness matrix, which is the appropriate formulation for the case of the free-edge effect and the matrix crack problem. The boundary stiffness matrix results from a Matrix-Riccati equation. The field quantities inside of the continuum can be calculated from an ordinary differential equation. 

For implementation of the model an object oriented approach based on the programming language C++ has been chosen. The authors developed the software library TOSCA (tools of object oriented software for continuum mechanics application) [8] which provides all classes required for description of continuum mechanics problems, as e.g. numerical solvers, discretization methods and data bases... 

An important factor to be considered is the computational cost. Continuum methods are noticeably less expensive than simulation methods based on discrete models. On the other hand, simple properties, as the solvation energy AGso of small and medium-size solutes are computed equally well with both continuum and discrete methods, reaching chemical accuracy (Orozco et al., 1992 Tomasi, 1994 Cramer and Truhlar, 1995a). There is large numerical evidence ensuring that the same conclusion holds for other continuum and discrete methods as well. The evaluation of A(jso at TS has given almost identical results in several cases, but here numerical evidence is not sufficient to draw definitive conclusions. 

The combination of all the points reported above seems to indicate versatile and efficient ab initio procedures as the best choice. However, there are other considerations to be added. Both continuum and discrete approaches suffer from limitations due to the separation of the whole liquid system into two parts, i.e. the primary part, or solute, and the secondary larger part, the solvent. These limitations cannot be eliminated until more holistic methods will be fully developed. We have already discussed some problems related to the shape of the cavity, which is the key point of this separation in continuum methods. We would like to remark that discrete methods suffer from similar problems of definition a tiny change in the non-boded interaction parameters in the solute-solvent interaction potential corresponds to a not so small change in the cavity shape. 

Accurate solvation procedures can be found in the family of continuum methods as well as in that of discrete methods. Now, the number of cases in which the application of methods belonging to the two families has given very similar results (with a good agreement with experimental data) is large. Continuum methods must take into account all the components of G, and they must use a realistic description of the cavity. Ad hoc parametriza-tions of cavities of simpler shapes such as to reproduce, for example, the desired value of an energy difference, often lead to considerable deformations of the reaction potential, and thus of the solute properties, which the interpretation of the phenomenon depends on. On the other side, discrete methods depend on the quality of the intermolecular potential as well as of the simulation procedure both are critical parameters. The simulation should also include the solvent electronic polarization, or some estimates of its effect. 

After finalizing the model equations and boundary conditions, the next task is to choose a suitable method to approximate the differential equations by a system of algebraic equations in terms of the variables at some discrete locations in space and time (called a discretization method). There are many such methods the most important are finite difference (FD), finite volume (FV) and finite element (FE) methods. Other methods, such as spectral methods, boundary element methods or cellular automata are used, but these are generally restricted to special classes of problems. All methods yield the same solution if the grid (number of discrete locations used to... 

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