Explicit and Implicit Methods

All of the methods mentioned so far are said to be explicit discretizations, since producing the next approximation point on a trajectory does not require solving any implicit equations defined in terms of the previous one. The Verlet method is not implicit even though Q appears on the right in the second equation, since it is given in an explicit way in terms of q and/ . Implicitness adds another layer in both analysis and numerical implementation, which, however, in certain applications, is readily justified. 

An implicit method will typically result in a system of nonlinear equations of the form 

Solving the system may then proceed, from an initial guess z +i by use of Newton s method  

The potentially demanding steps here are the calculation of the Jacobian matrix and the solution of the linear system. 

One way to save computational work is to recycle the Jacobian from a previous timestep. Alternatively, it may be possible to approximate the Jacobian crudely in such a way that the iteration stiU converges. For example, if the Jacobian matrix is large and can be written in the form 

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Solve with both the explicit and implicit methods the equation... 

The most common methodology when solving transient problems using the finite element method, is to perform the usual Garlerkin weighted residual formulation on the spatial derivatives, body forces and time derivative terms, and then using a finite difference scheme to approximate the time derivative. The development, techniques and limitations that we introduced in Chapter 8 will apply here. The time discretization, explicit and implicit methods, stability, numerical diffusion etc., have all been discussed in detail in that chapter. For a general partial differential equation, we can write... 

Both explicit and implicit methods are consistent and convergent. But, whereas the implicit method is absolutely stable, the explicit method is absolutely stable only if... 

Both the explicit and implicit methods already described have been improved to greater accuracy and, hopefully, greater efficiency. 

Such a treatment is not possible and one has to restore to approximations. We discussed some of those in section II. They could, as a first approximation, be split into three dilferent classes. In one set of approximations the solvent was treated within a simplified, parameterized approach. Thereby, the structure of the solvent was retained. As an alternative, the solvent could be approximated as a polarizable continuum that would respond to the presence of the solute but otherwise was without any internal structrue. These two types of approaches are coined explicit and implicit methods, respectively. Finally, a full quantum-mechanical treatment of either a smaller, finite supermolecule or an infinite, periodic supercell could be carried through. 

The explicit and implicit methods have their advantages and disadvantages, and one method is not necessarily better Ilian the other one. Next you will see that the explicit method is easy to implement but imposes a limit on the allowable time step to avoid instabilities in the solution, and the iinplicit method requires the nodal temperatures to be solved simultaneously for each time step but imposes no limit on the magnitude of the time step. We limit the discussion to. bne- and two-dimensional cases to keep the complexities at a manageable leyel, but the analysis can feadily be extended to threc-dimen.sional ca.ses and other coordinate systems. 

FIGURE 5-39 The formulation of explicit and implicit methods differs at the time step (previous or new) at wliich the heal transfer and heat generation terms are expre.ssed. 

Predictor-Corrector methods have been constructed attempting to combine the best properties of the explicit and implicit methods. The multistep methods are using information at more than two points. The additional points are ones at which data has already been computed. In one view, Adams methods arise from underlying quadrature formulas that use data outside of specifically approximate solutions computed prior to t . 

In the ADI method [4-6], the explicit and implicit methods are combined. The timesteps, AT, are divided into two half-timesteps, ATf2. For the first of these half-timesteps, the derivatives along the. -coordinate... 

In the next two sections, we discuss the basic theories underlying the essential differences between explicit and implicit methods. To help with this, we need to recall a few elementary steps such as the Newton interpolation formula. Details on the various interpolation theories can be found elsewhere (Burden and Faires 1981). 

To facilitate the development of explicit and implicit methods, it is necessary to briefly consider the origins of interpolation and quadrature formulas (i.e., numerical approximation to integration). There are essentially two methods for performing the differencing operation (as a means to approximate differentiation) one is the forward difference, and the other is the backward difference. Only the backward difference is of use in the development of eiqjlicit and implicit methods. 

A compromise between the explicit and implicit methods is the predictor-corrector technique, where the explicit method is used to obtain a first estimate of and this estimated y +i is then used in the RHS of the implicit formula. The result is the corrected y + i, which should be a better estimate to the true y +i than the first estimate. The corrector formula may be applied several times (i.e., successive substitution) until the convergence criterion (7.49) is achieved. Generally, predictor-corrector pairs are chosen such that they have truncation errors of approximately the same degree in h but with a difference in sign, so that the truncation errors compensate one another. 

EXPLICIT AND IMPLICIT METHODS USED IN REACTIVE FLOWS... 

In this expression, ( ) is the solution at time t and is the solution at time t + At. While certain flow conditions, such as compressible flow, are best suited to an explicit method for the solution of eq. (5-38), an impficit method is usually the most robust and stable choice for a wide variety of applications, including mixing. The major difference between the explicit and implicit methods is whether the right-hand side of eq. (5-38) is evaluated at the current time [F(( )) = F(ct))"] or at the new time [F(c )) = F(( )"+ )]. The implicit method uses the latter ... 

Both explicit and implicit methods have many variations. One of the differences between these methods is the order of the polynomial that is used for the approximation of the solution. The more sophisticated methods provide more accurate solutions, but the most important is to use a temporal and/or spatial stepsize that allows the stability of the method (Higham 1996). 

Implicit methods are favored for IVPs that are stiff, in which the condition number of the Jacobian matrix (the ratio of the largest and smallest eigenvalue moduli) is very large. Stiff systems are by no means rare and so we must be prepared to use both explicit and implicit methods. As a general rule, if we have no reason to expect that a problem is stiff, we first try an explicit method. If it fails, i.e., it keeps running with no end in sight, we try an imphcit method. A more detailed treatment of stiffness and a comparison of the numerical stability properties of exphcit and implicit methods are provided following the demonstration of the MATLAB ODE solvers. 

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