Numerical methods implicit

As an example, if only quasi-steady flow elements are used with volume pressure elements, a model s smallest volume size (for equal flows) will define the timescale of interest. Thus, if the modeler inserts a volume pressure element that has a timescale of one second, the modeler is implying that events which happen on this timescale are important. A set of differential equations and their solution are considered stiff or rigid when the final approach to the steady-state solution is rapid, compared to the entire transient period. In part, numerical aspects of the model will determine this, but also the size of the perturbation will have a significant impact on the stiffness of the problem. It is well known that implicit numerical methods are better suited towards solving a stiff problem. (Note, however, that The Mathwork s software for real-time hardware applications, Real-Time Workshop , requires an explicit method presumably in order to better guarantee consistent solution times.)... 

Unsteady-state conduction and implicit numerical method. In some practical problems the restrictions imposed on the. value M >2 hy stability requirements may prove inconvenient. Also, to minimize the stability problems, implicit methods using different finite-difference formulas have been developed. An important one of these formulas is the... 

Other cases such as for diffusion between dissimilar slabs in series, resistance between slabs in series, and so on, are covered in detail elsewhere (Gl), with actual numerical examples being given. Also, in reference (Gl) the implicit numerical method is discussed. 

There are explicit and implicit numerical methods for the finite-difference version of Eqs. 9.27 and 9.28. There are also approximate numerical methods in which the radial derivatives are replaced by the functions obtained by differentiating assumed trial functions for the radial profiles, essentially transforming the equations into ordinary differential equations in z. The cell model (Hlavacek and Votruba, 1977) can also be used for the solution. 

Z/v weighting factor for implicit numerical method in Table... 

I hese equations cannot be used directly, and numerical methods are needed to compute the velocity components. The velocity components can be found by implicit differentiation and using an iterative technique.-" ... 

Within esqjlicit schemes the computational effort to obtain the solution at the new time step is very small the main effort lies in a multiplication of the old solution vector with the coeflicient matrix. In contrast, implicit schemes require the solution of an algebraic system of equations to obtain the new solution vector. However, the major disadvantage of explicit schemes is their instability [84]. The term stability is defined via the behavior of the numerical solution for t —> . A numerical method is regarded as stable if the approximate solution remains bounded for t —> oo, given that the exact solution is also bounded. Explicit time-step schemes tend to become unstable when the time step size exceeds a certain value (an example of a stability limit for PDE solvers is the von-Neumann criterion [85]). In contrast, implicit methods are usually stable. 

This then provides a physical derivation of the finite-difference technique and shows how the solution to the differential equations can be propagated forward in time from a knowledge of the concentration profile at a series of mesh points. Algebraic derivations of the finite-difference equations can be found in most textbooks on numerical analysis. There are a variety of finite-difference approximations ranging from the fully explicit method (illustrated above) via Crank-Nicolson and other weighted implicit forward. schemes to the fully implicit backward method, which can be u.sed to solve the equations. The methods tend to increase in stability and accuracy in the order given. The difference scheme for the cylindrical geometry appropriate for a root is... 

The computational code used in solving the hydrodynamic equation is developed based on the CFDLIB, a finite-volume hydro-code using a common data structure and a common numerical method (Kashiwa et al., 1994). An explicit time-marching, cell-centered Implicit Continuous-fluid Eulerian (ICE) numerical technique is employed to solve the governing equations (Amsden and Harlow, 1968). The computation cycle is split to two distinct phases a Lagrangian phase and a remapping phase, in which the Arbitrary Lagrangian Eulerian (ALE) technique is applied to support the arbitrary mesh motion with fluid flow. 

Though this new algorithm still requires some time step refinement for computations with highly inelastic particles, it turns out that most computations can be carried out with acceptable time steps of 10 5 s or larger. An alternative numerical method that is also based on the compressibility of the dispersed particulate phase is presented by Laux (1998). In this so-called compressible disperse-phase method the shear stresses in the momentum equations are implicitly taken into account, which further enhances the stability of the code in the quasi-static state near minimum fluidization, especially when frictional shear is taken into account. In theory, the stability of the numerical solution method can be further enhanced by fully implicit discretization and simultaneous solution of all governing equations. This latter is however not expected to result in faster solution of the TFM equations since the numerical efforts per time step increase. 

The successful numerical solution of differential equations requires attention to two issues associated with error control through time step selection. One is accuracy and the other is stability. Accuracy requires a time step that is sufficiently small so that the numerical solution is close to the true solution. Numerical methods usually measure the accuracy in terms of the local truncation error, which depends on the details of the method and the time step. Stability requires a time step that is sufficiently small that numerical errors are damped, and not amplified. A problem is called stiff when the time step required to maintain stability is much smaller than that that would be required to deliver accuracy, if stability were not an issue. Generally speaking, implicit methods have very much better stability properties than explicit methods, and thus are much better suited to solving stiff problems. Since most chemical kinetic problems are stiff, implicit methods are usually the method of choice. 

Integration of a time-dependent thermal-capillary model for CZ growth (150, 152) also has illuminated the idea of dynamic stability. Derby and Brown (150) first constructed a time-dependent TCM that included the transients associated with conduction in each phase, the evolution of the crystal shape in time, and the decrease in the melt level caused by the conservation of volume. However, the model idealized radiation to be to a uniform ambient. The technique for implicit numerical integration of the transient model was built around the finite-element-Newton method used for the QSSM. Linear and nonlinear stability calculations for the solutions of the QSSM (if the batchwise transient is neglected) showed that the CZ method is dynamically stable small perturbations in the system at fixed operating parameters decayed with time, and changes in the parameters caused the process to evolve to the expected new solutions of the QSSM. The stability of the CZ process has been verified experimentally, at least... 

In the context of implicit solvent models, several numerical methods based on integral equations (DPCM, COSMO, IEF,...) have been proposed for calculating reaction potentials and energies. 

The numerical solution to the advection-dispersion equation and associated adsorption equations can be performed using finite difference schemes, either in their implicit and/or explicit form. In the one-dimensional MRTM model (Selim et al., 1990), the Crank-Nicholson algorithm was applied to solve the governing equations of the chemical transport and retention in soils. The web-based simulation system for the one-dimensional MRTM model is detailed in Zeng et al. (2002). The alternating direction-implicit (ADI) method is used here to solve the three-dimensional models. 

For systems involving recycle streams or intermediate feed locations, the method of successive substitution can be used [Mohan and Govind, 1988a]. Moreover, multiple reactions including side reactions and series, parallel or series-parallel reactions result in strongly coupled differential equations. They have been solved numerically using an implicit Euler method [Bernstein and Lund, 1993]. 

Implicit solvation models developed for condensed phases represent the solvent by a continuous electric field, and are based on the Poisson equation, which is valid when a surrounding dielectric medium responds linearly to the charge distribution of the solute. The Poisson equation is actually a special case of the Poisson-Boltzmann (PB) equation PB electrostatics applies when electrolytes are present in solution while the Poisson equation applies when no ions are present. Solving the Poisson equation for an arbitrary equation requires numerical methods, and many researchers have developed an alternative way to approximate the Poisson equation that can be solved analytically, known as the... 

Equation 15 is an implicit function of in terms of measurable quantities and the four membrane parameters L, gr, qj, and K, and it can be solved by numerical methods. Substituting the values of into Equation 14 yields Cj, which in turn allows to be calculated from Equation 10. Then the quantities V and c can be calculated using Equations 7 and 8. perm perm... 

The opposite viewpoint, advocated by Boris et al. [16], is that no explicit filtering should be performed and no explicit residual stress model should be used (Oj = 0). Instead, an appropriate numerical method is used to attempt to solve the Navier-Stokes equation for v(r, f). Because the grid is not fine enough to resolve the solution to the Navier-Stokes equation, significant numerical stresses arise. Thus, filtering and residual-stress modeling are performed implicitly by the numerical method. 

An unsophisticated numerical method is used here for simplicity. Various implicit and explicit methods of writing dilference equations and their limitations are discussed by Leon Lapidus in Digital Computation for Chemical Engineers, McGraw-Hill Book Company, New York, 1960. The Crank-Nicholson implicit method [G. Crank and P. Nicholson, Proc. Cambridge Phil. Soc., 43, 50 (1947)] is well suited for machine computation. 

The coupled set of nonlinear differential equations (equations 1 and 4) are solved by the alternating direction implicit (ADI) method f9-10) on an evenly spaced grid. The advective transport of a solute species was solved using the Lax-Wendroff two-step method (10). To ensure that numerical dispersion is avoided, a grid spacing was chosen such that the grid Peclet number (defined by < 2 fll). The computational expense involved in using a... 

Equation (1-13) consists of one equation in one unknown, the temperature. The form of the implicit function X,(T) generally requires that the solution of Eq. (1-13) for the bubble-point temperature be effected by a trial-and-error procedure. Of the many numerical methods for solving such a problem, only Newton s method2,5 is presented. In the application of this method, it is convenient to restate Eq. (1-13) in functional forip as follows... 

A few examples will be demonstrated in the following section. Each feature was incorporated in the software because it has been found necessary or useful in some modelling project. All the modelling tasks that previously required tailor made solutions in each project can now be solved in a unified manner in the ModEst environment. ModEst is able to deal with explicit algebraic, implicit algebraic (systems of nonlinear equations) and ordinary differential equations. As the standard way to handle PDE systems, the Numerical Method of Lines, which transforms a PDE system to a number of ODE components is used. In addition, any model with a solver provided may be dealt with as an algebraic system. The basic numerical tools are contained in the well tested public domain software (Bias, Linpack, Eispack, LSODE). 

Table 8 Properties of numerical methods with non-empty interval of periodicity. A.O. is the algebraic order of the method. P.L.O. is the phase-lag order, Int. Per. is the interval of periodicity of the method. I/E is an indication for implicit or explicit methods...

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