Discretization schemes

G. Zhang and T. Schlick. Implicit discretization schemes for Langevin dynamics. Mol. Phys., 84 1077-1098, 1995. 

Hence, as the second class of techniques, we discuss adaptive methods for accurate short-term integration (Sec. 4). For this class, it is the major requirement that the discretization allows for the stepsize to adapt to the classical motion and the coupling between the classical and the quantum mechanical subsystem. This means, that we are interested in discretization schemes which avoid stepsize restrictions due to the fast oscillations in the quantum part. We can meet this requirement by applying techniques recently developed for evaluating matrix exponentials iteratively [12]. This approach yields an adaptive Verlet-based exponential integrator for QCMD. 

It should also be remembered that the discretization scheme influences the accuracy of the results. In most CFD codes, different discretization schemes can be chosen for the convective terms. Usually, one can choose between first-order schemes (e.g., the first-order upwind scheme or the hybrid scheme) or second-order schemes (e.g., a second-order upwind scheme or some modified QUICK scheme). Second-order schemes are, as the name implies, more accurate than first-order schemes. However, it should also be remembered that the second-order schemes are numerically more unstable than the first-order schemes. Usually, it is a good idea to start the computations using a first-order scheme. Then, when a converged solution has been obtained, the user can continue the calculations with a second-order scheme. 

Hounslow etal. (1988), Hounslow (1990a), Hostomsky and Jones (1991), Lister etal. (1995), Hill and Ng (1995) and Kumar and Ramkrishna (1996a,b) present numerical discretization schemes for solution of the population balance and compute correction factors in order to preserve total mass and number whilst Wojcik and Jones (1998a) evaluated various methods. 

Equation (8.29) provides no guarantee of stability. It is a necessary condition for stability that is imposed by the discretization scheme. Practical experience indicates that it is usually a sufficient condition as well, but exceptions exist when reaction rates (or heat-generation rates) become very high, as in regions near thermal runaway. There is a second, physical stability criterion that prevents excessively large changes in concentration or temperature. For example. An, the calculated change in the concentration of a component that is consumed by the reaction, must be smaller than a itself Thus, there are two stability conditions imposed on Az numerical stability and physical stability. Violations of either stability criterion are usually easy to detect. The calculation blows up. Example 8.8 shows what happens when the numerical stability limit is violated. 

This condition can easily be verified in the case of discrete schemes for equations of mathematical physics. It allows one to extract from the primary family of schemes the set of stable schemes, within which one should look for schemes with a prescribed accuracy, volume of computations, and other desirable properties and parameters. 

The discretization of the dilation parameter need not be dyadic. Discretization schemes with integer or noninteger factors are possible and have been suggested. The reconstruction of Fit) is given by... 

Unfortunately, the requirements for translational invariance of the wavelet decomposition are difficult to satisfy. Consequently, for either discretization scheme, comparison of the wavelet coefficients for two signals may mislead us into thinking that the two trends are different, when in fact one is simply a translation of the other. 

In computational fluid dynamics only the last two methods have been extensively implemented into commercial flow solvers. Especially for CFD problems the FVM has proven robust and stable, and as a conservative discretization scheme it has some built-in mechanism of error avoidance. For this reason, many of the leading commercially available CFD tools, such as CFX4/5, Fluent and Star-CD, are based on the FVM. The oufline on CFD given in this book wiU be based on this method however, certain parts of the discussion also apply to the other two methods. 

When the transport equation for c is solved with a discretization scheme such as upwind, artificial diffusive fluxes are induced, effecting a smearing of the interface. When these diffusive fluxes are significant on the time-scale of the simulation, the information on the location of different fluid volumes is lost. The use of higher order discretization schemes is usually not sufficient to reduce the artificial smearing of the interface to a tolerable level. Hence special methods are used to guarantee that a physically reasonable distribution of the volume fraction field is maintained. 

If for example we discretize the region over which the PDE is to be solved into M grid blocks, use of finite differences (or any other discretization scheme) to approximate the spatial derivatives in Equation 10.1 yields the following system of ODEs ... 

More basically, LB with its collision rules is intrinsically simpler than most FV schemes, since the LB equation is a fully explicit first-order discretized scheme (though second-order accurate in space and time), while temporal discretization in FV often exploits the Crank-Nicolson or some other mixed (i.e., implicit) scheme (see, e.g., Patankar, 1980) and the numerical accuracy in FV provided by first-order approximations is usually insufficient (Abbott and Basco, 1989). Note that fully explicit means that the value of any variable at a particular moment in time is calculated from the values of variables at the previous moment in time only this calculation is much simpler than that with any other implicit scheme. 

Instances of a task are replicas of the task operating under different conditions. The concept is used to optimize the operating conditions, such as the column pressure, and assumes the development of an operating range and a discretization scheme. Feasible ranges of pressure are identified by the physical properties (e.g., critical pressure) of the key components (upper limit) and the available utility levels (lower limit). The discretization scheme may be either uniform or based on the available utilities. The modeler can use a small or large number of discrete levels to capture associated trade-offs. 

There are couple of measures that can taken in order to minimize ND [37]. Higher order discretization schemes such as the QUICK scheme reduce the numerical errors. Furthermore, ND depends strongly on the relative orientation of flow velocity and grid cells. ND can be minimized by choosing grid cells with edges parallel to the local flow velocity. 

Figure8-1 Space-time grid for the one-dimensional diffusion equation, evidencing the explicit forward-difference, implicit backward-difference and C rank-Nicholson discretization schemes.

To obtain an algorithm that is unconditionally stable, we consider an implicit discretization scheme that results from using backward finite-differences for the time derivative. The corresponding difference equation is most conveniently obtained by approximating the diffusion equation at point (Xj,tn+i) ... 

Because of the success encountered by finite elements in the solution of elliptic problems, it was extended (in the 80s) to the advection or transport equation which is a hyperbolic equation with only one real characteristic. This equation can be solved naturally for an analytical velocity field by solving a time differential equation. It appeared important, when the velocity field was numerically obtained, to be able to solve simultaneously propagation and diffusion equations at low cost. By introducing upwinding in test functions or in the discretization scheme, the particular nature of the transport equation was considered. In this case, a particular direction is given at each point (the direction of the convecting flow) and boundary conditions are only considered on the part of the boundary where the flow is entrant. 

The numerical procedure requires approximating the unknowns in the simple computational domain defined by a stream band B in the mapped domain D. The stream bands are divided into rectangular elements built on two rectilinear streamlines. Meshes are generally refined in the vidnity of contraction sections of the flow domains, as is usually the case. The discretization schemes adopted depend on the problem under consideration. We find it of interest to underline the following points ... 

The moderate and low Peclet regimes are characterized by significant magnitudes of all the terms of eq. (2), which should therefore be solved numerically. A non-uniform finite -difference discretization scheme has been chosen for the system, estimating the overall adsorption efficiency, Xg, as follows ... 

The above division leads to the following discrete scheme ... 

A discrete scheme (3) is a symplectic scheme if the transformation matrix S is symplectic. 

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