Discrete equation

These equations are the coupled system of discrete equations that define the rigorous forward problem. Note that we can take advantage of the convolution form for indices (i — I) and (j — J). Then, by exciting the conductive media with a number N/ oi frequencies, one can obtain the multifrequency model. The kernels of the integral equations are described in [13] and [3j. 

In practice, in order to maintain the symmetry of elemental coefficient matrices, some of the first order derivatives in the discretized equations may also be integrated by parts. 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

In Equation (4.12) the discretization of velocity and pressure is based on different shape functions (i.e. NjJ = l,n and Mil= l,m where, in general, mweight function used in the continuity equation is selected as -Mi to retain the symmetry of the discretized equations. After application of Green s theorem to the second-order velocity derivatives (to reduce inter-element continuity requirement) and the pressure terms (to maintain the consistency of the formulation) and algebraic manipulations the working equations of the U-V-P scheme are obtained as... 

Solution of the algebraic equations. For creeping flows, the algebraic equations are hnear and a linear matrix equation is to be solved. Both direct and iterative solvers have been used. For most flows, the nonlinear inertial terms in the momentum equation are important and the algebraic discretized equations are therefore nonlinear. Solution yields the nodal values of the unknowns. 

The governing equations, (9.1)-(9.4), are approximated with discrete equations on the computational mesh. The discrete equations can be derived... 

Schemes with weights. When discretizing equation (2) in t, the scheme with weights arises naturally in one or another form ... 

Third, writing the discretized equations in matrix form results in sparse matrices, often of a tri-diagonal form, which traditionally are solved by successive under- or over-relaxation methods using the tri-diagonal matrix algorithm... 

The discretized equations of the finite volume method are solved through an iterative process. This can sometimes have difficulty converging, especially when the nonlinear terms play a strong role or when turbulence-related quantities such as k and s are changing rapidly, such as near a solid surface. To assist in convergence a relaxation factor can be introduced ... 

To determine when a solution is converged usually involves examining the residual values. The residual value is a measure of the imbalance in the discretized equation, summed over all the computational cells in the domain. Residuals can be obtained for continuity, velocity components, and turbulence variables. Again, it is common practice to set a cut-off value for the normalized residual values. When the set value is reached, the iteration process is stopped. Our experience with packed-tube simulations, especially if low... 

To discretize equation (2.1), we will identify elements along the z-coordinate as k and elements along the time coordinate as n. Then, the individual terms become... 

To solve the equation by simulation, we replace the discrete equation by the following Markov process. A number of particles is distributed in... 

This system of equations may be solved numerically. In finite-difference form on a uniform mesh (1 < j < J), the discrete equations are stated as... 

A second way consists in calculating the derivatives (d/dXi)E p(p, p ) of the approximated energy Efp(p,p ). This second approach can be subdivided into three methods (d/d i)E s>(p, p ) can be computed (i) by finite differences, (ii) by deriving analytically the discrete equations used for the calculation of E p, p ), (iii) by automatic differentiation [24]. Although (ii) and (iii) are theoretically equivalent, they are not in practice they correspond to two dramatically different implementations of a single mathematical formalism. 

For the mathematics of this, consider the discrete equation resulting from the Euler method, as in (4.8). Note that the new point, yn+i is formed from the old point yn by the addition of a term, here St f(yn). With RK, these terms are given the symbols fcj, there are from one to several of them, and they are added in a weighted manner. The procedure is to generate a number of these k s. One begins with an Euler step,... 

If several species are involved (in this case there is the product prod, but we are not interested in it), the equations are extended in an obvious manner, apart from some tricks to be seen in a later chapter in connection with implicit methods. This is one of the attractive aspects of method EX. If the her is second order, there will be a term in C in the discrete equation, and it will present no problem in the discretisation step [146]. 

We begin with the simpler case of the two species not being coupled, that is, each of their discrete equations contains only terms from one of the species. The coupled case is given below, being rather more complicated. There is of course coupling of the two species by the boundary conditions. 

Up to this point, the treatments have involved reactions for which the discrete form of the reaction-diffusion equations involve only terms in concentration of the species to which the discrete equation applies. That is, if there were two substances involved, O and R as above, then the discrete equation at a point i had terms only in C 0 for species O, and only C R for species R. This made it possible to use the Thomas algorithm to reduce a system like (6.27) to (6.28), treating the two species systems separately. They then get coupled through the boundary conditions. 

When homogeneous reactions take place, it often happens that some of the discrete equations contain terms in concentration for more than the one species, and it is then not generally possible to use the Thomas algorithm to reduce the systems. These systems are said to be coupled. An example will illustrate this situation. 

The derivation of the discrete equations corresponding to this reaction pair will be given in Chap. 8 and it will suffice here to provide the general form they will take ... 

Coupled equations are those in which some or all of the dynamic equations have terms in more than one of the variables (concentrations). This leads, upou discretisation, to systems of discrete equations that cannot usually be solved using the Thomas algorithm because, no matter how one orders the concentration vectors, the systems correspond to matrix equations that are... 

The stability of the Saul yev schemes in the electrochemical context with mixed boundary conditions, was examined [112, 118]. Surprisingly, it was found that the LR variant can be unstable with mixed boundary conditions. There exists, for any number N of intervals in space, a maximum A value in the discrete equation, above which the LR scheme becomes unstable. Fortunately, it is rather difficult to attain this condition in practice. Since these studies, Deng [207] has used the various Saul yev schemes and offshoots, and cites several Chinese studies also using Saul yev variants, but they have found little application elsewhere. 

Repeating this for the method BI, where the discrete equation at point i is... 

The vector can be calculated either with the normal Kalman Filter (KF) which gives Xk for the discrete equation state (F(Xk, Uk, Vk)) or with the extended Kalman filter (KFE) which gives Pk+i in the calculation system. For this estimation, it is also necessary to obtain the state of the system Xk from the next state Xk+j. This estimation is made by block IT (inversion translator) another IT block gives... 

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