Explicit methods

A reasonable approach for achieving long timesteps is to use implicit schemes [38]. These methods are designed specifically for problems with disparate timescales where explicit methods do not usually perform well, such as chemical reactions [39]. The integration formulas of implicit methods are designed to increase the range of stability for the difference equation. The experience with implicit methods in the context of biomolecular dynamics has not been extensive and rather disappointing (e.g., [40, 41]), for reasons discussed below. 

Another option is a q,p) = p and b q,p) = VU q). This guarantees that we are discretizing a pure index-2 DAE for which A is well-defined. But for this choice we observed severe difficulties with Newton s method, where a step-size smaller even than what is required by explicit methods is needed to obtain convergence. In fact, it can be shown that when the linear harmonic oscillator is cast into such a projected DAE, the linearized problem can easily become unstable for k > . Another way is to check the conditions of the Newton-Kantorovich Theorem, which guarantees convergence of the Newton method. These conditions are also found to be satisfied only for a very small step size k, if is small. 

S. K. Gray, D. W. Noid and B. G. Sumpter, Symplectic integrators for large scale molecular dynamics simulations A comparison of several explicit methods , J. Chem. Phys., Vol 101, no 5, 4062-72, 1994. 

Runge-Kutta methods are explicit methods that use several function evaluations for each time step. Runge-Kutta methods are traditionally written for/(f, y). The first-order Runge-Kutta method is Euler s method. A second-order Runge-Kutta method is... 

Repeat steps 2 through 6 with a corrector step for the same time increment. Repeat again for any further predictor and/or predictor-corrector steps that may be advisable. Distefano (ibid.) discusses and compares a number of suitable explicit methods. 

The main goal of subsequent considerations is the comparison between ADM of the type (17) with parameters (25) and the explicit method with optimal set of Chebyshev s parameters... 

The method of steepest descent. The explicit method of steepest descent is given by the formulas... 

Equilibrium data correlations can be extremely complex, especially when related to non-ideal multicomponent mixtures, and in order to handle such real life complex simulations, a commercial dynamic simulator with access to a physical property data-base often becomes essential. The approach in this text, is based, however, on the basic concepts of ideal behaviour, as expressed by Henry s law for gas absorption, the use of constant relative volatility values for distillation and constant distribution coeficients for solvent extraction. These have the advantage that they normally enable an explicit method of solution and avoid the more cumbersome iterative types of procedure, which would otherwise be required. Simulation examples in which more complex forms of equilibria are employed are STEAM and BUBBLE. 

Variable step, 5th-order, Runge-Kutta explicit method (ALGO = 0). 

Especially for the electrons, the fluid model has the advantage of a lower computational effort than the PIC/MC method. Their low mass (high values of the transport coefficients) and consequent high velocities give rise to small time steps in the numerical simulation (uAf < Aa) if a so-called explicit method is used. This restriction is easily eliminated within the fluid model by use of an implicit method. Also, the electron density is strongly coupled with the electric field, which results in numerical Instabilities. This requires a simultaneous implicit solution of the Poisson equation for the electric field and the transport equation for the electron density. This solution can be deployed within the fluid model and gives a considerable reduction of computational effort as compared to a nonsi-multaneous solution procedure [179]. Within the PIC method, only fully explicit methods can be applied. 

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... 

In Equation (23), all input data are known from the earlier time step. This way of calculating the values for the next time step is called explicit method. More information on numerical modeling, and for example how to integrate convection, can be found in Farlow (1983) and Ozisik (1968). 

The explicit methods avoid the need of solving large sets of equations and can therefore be used on smaller computers. However, these methods tend to be unstable unless the step sizes are kept small and an artificial constraint is introduced on the variables. In most formulations the time step must be less than the reach length divided by the isothermal sonic velocity (see Section V,B,1). Explicit methods are used in PIPETRAN (D7) and SATAN (G4). 

The stability limits for the explicit methods are based on the largest eigenvalue of the linearized system of equations... 

The price of using implicit methods is that one now has a system of equations to solve at each time step, and the solution methods are more complicated (particularly for nonlinear problems) than the straightforward explicit methods. Phenomena that happen quickly can also be obliterated or smoothed over by using a large time step, so implicit methods are not suitable in all cases. The engineer must decide if he or she wants to track those fast phenomena, and choose an appropriate method that handles the time scales that are important in the problem. 

The first three methods can be considered more or less explicit methods for the determination of process components and parameters. The four groups of methods for investigation of anaerobic transformations will be outlined in the following. 

Linearizing the kinetic term as before, a set of three unknown linear equations is obtained, which is completed by the finite difference expression of the initial and boundary conditions. Inversion of the ensuing matrix allows the calculation of C at each node of the calculation grid and finally, of the current flowing through the electrode, or of the corresponding dimensionless function, by means of its finite difference expression. Calculation inside thin reaction layers may thus be more efficiently carried out than with explicit methods. The combination of the Crank-Nicholson... 

As discussed in the introduction to this chapter, the solution of ordinary differential equations (ODEs) on a digital computer involves numerical integration. We will present several of the simplest and most popular numerical-integration algorithms. In Sec, 4.4.1 we will discuss explicit methods and in Sec. 4.4.2 we will briefly describe implicit algorithms. The differences between the two types and their advantages and disadvantages will be discussed. 

The explicit methods considered in the previous section involved derivative evaluations, followed by explicit calculation of new values for variables at the next point in time. As the name implies, implicit integration methods use algorithms that result in implicit equations that must be solved for the new values at the next time step. A single-ODE example illustrates the idea. 

The main advantage of the implicit algorithms is that they do not become numerically unstable. Very large step sizes can be taken without having to worry about the instability problems that plague the explicit methods. Thus, the implicit methods are very useful for stiff systems. 

Indole. Russian patent 306, 126 (1971). Note See the making precursors section of this book for an explicit method of producing methylindole in good yield. Add 26 ml of 0.77 N NaOCl to a stirred suspension of 1.9 g of orthocarbamoyl cinnamamide in 50 ml of methanol and heat in a distillation apparatus until no more indoles is distilled off (use an indole test) or just heat at 40° for 2 hours. Extract the distillate with chloroform, dry with most any drying agent, except calcium chloride, and evaporate the solvent (CHCb) off from the remaining indole. Yield 45%. 

Modiiying the Scheil Solidification Model 11.3.3.3 More Explicit Methods of Accounting for 447... 

More Explicit Methods of Accounting for Back-Difhision... 

Figure 3-14 Schematics of dividing the diffusion medium into N equally spaced divisions. Starting from the initial condition (concentration at every nodes at f = 0), C of the interior node at the next time step (f = At) can be calculated using the explicit method, whereas C at the two ends can be obtained from the boundary condition.

Because the diffusion profiles at the next time step are calculated directly from initial and boundary conditions, this method is called the explicit method. The method is stable only when a <0.5, i.e., Ax/(A ) < 0.5, and has only first-order precision because the expression for (dCldx) has only first-order precision. Hence, given A, it is necessary to choose a small Ax. 

If D depends on concentration, the explicit method is easy to adapt but the implicit methods are more difficult. Let... 

In an explicit method, the spatial derivatives are evaluated based on the dependent variables at time level n, that is, w". Thus, for the problem at hand,... 

The explicit method is implemented easily in a computer program or a spreadsheet. Unfortunately, there is an issue of stability that must be considered. It turns out that when the coefficient of wn-, the term in square brackets, becomes negative, the solution becomes unstable and the approach is completely unusable. It is evident by inspection that if the... 

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