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Implicit techniques

Implicit integration techniques involve various interpolation formulas that include the y term. Some of the techniques are described in the following sections. [Pg.254]

Runge-Kutta-Fehlberg (RKF) is another method involving fourth- and fifth-order integration formulations (Equations 11.11 and 11.12, respectively) whose difference can be [Pg.254]

These formulations require evaluation of six functions per step  [Pg.254]

The method is second order that is given with the following equation  [Pg.254]

The method involves the use of the following formulas that vary with the orders  [Pg.254]


Contents Introduction. - Basic Equations. -Diffusional Transport - Digitally. - Handling of Boundary Problems. - Implicit Techniques and Other Complications. - Accuracy and Choice. -Non-Diffusional Concentration Changes. - The Laplace Equation and Other Steady-State Systems. - Programming Examples. - Index. [Pg.120]

Unlike explicit methods, the performance of implicit methods cannot be simply judged by conventional statistical measures such as goodness of fit. As pointed out in the literature,18 spurious effects such as system drift and covariations among constituents can be incorrectly interpreted as arising from the analyte of interest. This scenario has led to the development of hybrid methods in which elements of explicit and implicit techniques are combined to improve performance. [Pg.337]

As an alternative to the simultaneous solution of stiff differential equations through an implicit technique a method is described here which approximates the solution by successive computations of the corresponding finite difference equations. The successive nature of this method essentially decouples the K(N + 1)... [Pg.217]

Briley, W. R., and McDonald, H., Solution of the three-dimensional compressible Navier-Stokes equations by an implicit technique. Proc. 4th Int. Cortf. Num. Methods in Fluid Dyruimics, Lecture Notes in Physics, Springer-Verlag, Berlin, 1975, vol. 35, p. 105. [Pg.320]

Now, even if values of c" i = 1,2,..., ) are known, (25.95) cannot be solved explicitly, as c"+l is also a function of the unknown c, 1 and c" /. However, all k equations of the form (25.95) for i = 1,2,..., k form a system of linear algebraic equations with k unknowns, namely, c"+l, c"+l,..., c"+l. This system can be solved and the solution can be advanced from r to tn+. This is an example of an implicit finite difference method. In general, implicit techniques have better stability properties than explicit methods. They are often unconditionally stable and any choice of Ar and Ax may be used (the choice is ultimately based on accuracy considerations alone). [Pg.1118]

Numerical solution of the ODEs describing initial value problems is possible by explicit and implicit techniques, which are described in the Sections 11.1.1 and 11.1.2, respectively. It is worth noting that the techniques are formulated for the solution of a single equation (Equation 11.1), but they can be used for solving multiple ODEs as well. Theoretical background of these methods as well as their stabilities are described elsewhere [1,2] and will not be discussed here. [Pg.253]

The main advantage of the implicit techniques is their stability for any given value of the step size. This advantage, however, requires the solution of a set of nonlinear equations via an iterative approach. For this purpose, methods such as successive substitution or Newton-Raphson can be used [1]. [Pg.255]

The boundary conditions that may be used with the thermochemical module include specified boundary temperature, convective heat transfer or no heat transfer (adiabatic). Different conditions (e.g., different HTCs) can be applied to each element as desired. Either explicit or implicit techniques may be chosen to solve the heat transfer (Eq. [13.1]) and cure rate equations. Using either technique, these two equations are uncoupled during each solution time-step. This approach facilitates a simplified and modular solution procedure and is sufficiently accurate if small time steps are used. [Pg.419]

In order to deal with the case of space-dependent dielectric constant in multidimensional space, alternating direction implicit techniques [82] are developed after revwiting Eq. (6.102) as... [Pg.316]

Implicit formulas of the type described above have been found to be unconditionally stable. It can be generalized that most explicit finite difference approximations are conditionally stable, whereas most implicit approximations are unconditionally stable. The explicit methods, however, are computationally easier to solve than the implicit techniques. [Pg.401]


See other pages where Implicit techniques is mentioned: [Pg.336]    [Pg.316]    [Pg.191]    [Pg.316]    [Pg.217]    [Pg.74]    [Pg.148]    [Pg.1220]    [Pg.965]    [Pg.254]    [Pg.169]    [Pg.310]    [Pg.316]   


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Implicit

Implicit integration techniques

Implicit techniques ADI

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