Explicit algorithms

Compare this carefully with the explicit algorithm given in Eq. (4.47). The derivative is evaluated at the next step in time where we do not know the variable i. Thus, the unknown appears on both sides of the equation. Consider the simple ODE... 

It is clear from this expression that the method is stable for all A. and all time steps hn that is, the method is unconditionally stable (for linear problems). A consequence of the strong stability is that the time step can be chosen primarily to maintain accuracy. In the slowly varying regions of stiff problems, the time steps can be very large compared those required to maintain stability for an explicit algorithm. 

Alt and Ceschino [167] have established a simply explicit algorithm, A-stable and of order three, according to the scheme... 

S is nothing other than the stiffness ratio of the system. The number of steps for Euler s explicit algorithm increases with the stiffness of the system. This conclusion is quite general or, in other words, explicit algorithms are not suitable for integrating stiff systems. 

Consider flow over a flat plate. The computation is started by assuming that uy = u,y, at the leading edge and v, = 0. The value of vy is needed in the explicit algorithm to move on to the i+1 level. It is not required to specify the initial values of vy in the formal mathematical formulation of the partial differential equation. A suitable initial distribution for vy can be obtained by using the continuity equation to eliminate du/5x from the x-momentum equation. For a laminar, incompressible flow, this means that... 

In ref. 144 the author presents the construction of a non-standard explicit algorithm for initial-value problems. The order of the developed method is two and also is A-stable. The new proposed method is proven to be suitable for solving different kind of initial-value problems such as non-singular problems, singular problems, stiff problems and singularly perturbed problems. Some numerical experiments are considered in order to check the behaviour of the method when applied to a variety of initial-value problems. 

The wavefunction operator (WFO) approach of Luzanov, Wulfov, and Krouglov (1992)224 seems to be the same as Harrison s CI+PT approach, although it is formulated differently and implemented using determinants. This method appears to involve the same amount of work as other determinant-based sparse Cl methods,215,220,221 but it uses rather different intermediate arrays. Explicit algorithms for the WFO method were presented in 1996 by Wulfov,225 who obtained results for HF dimer in a 4s3pld/2slp basis using a... 

In the present context, this example was intended to serve as a reminder of how one formulates a simple model for the quantum mechanics of electrons in metals and, also, how the Pauli principle leads to an explicit algorithm for the filling up of these energy levels in the case of multielectron systems. In addition, we have seen how this model allows for the explicit determination (in a model sense) of the cohesive energy and bulk modulus of metals. 

Stability is a problem associated with most explicit advection algorithms. The explicit algorithm of (25.130) has the worst behavior. The upwind scheme is stable if... 

The following classes for the solution of nonstiff problems, which are based on Runge-Kutta explicit algorithms, are implemented in the BzzMath library ... 

All the explicit algorithms are unstable and, therefore, the implicit ones ( ) must be used. Consequently, a nonlinear system has to be solved at each iteration. 

The predictor-corrector method takes its name from an application of an explicit algorithm (predictor) followed by an implicit algorithm (corrector of the prevision). 

If the iterations are not fully accomphshed, the system (2.225) is not accurately solved and the resulting algorithm is an explicit algorithm (even though computationally heavy), which does not have the stabihty features of the implicit method used in the iterations. 

While in the case of ODE with initial conditions k of the explicit algorithms do not require any nonlinear system solution, this is no longer true for BVP. It is therefore reasonable to adopt implicit algorithms that are preferable in terms of stabiUty. 

The prosodic content is generated by explicit algorithms, and signal processing techniques are used to modify the pitch and timing of the diphones to match that of the specification. 

In this review, we shall restrict ourselves to considering the problem of calculating the potential energy for the system and shall not discuss explicitly algorithms, such as molecular dynamics or Monte Carlo, which employ the energy... 

Blencoe (in press). This paper reviews the characteristics of the two-parameter Margules, van Laar and quasi-chemical solution models for binary mixtures, and presents explicit algorithms in the form of F0RTRAN IV subroutines for calculating the adjustable coefficients for the several models from (experimental) data on the compositions and equilibration temperatures of two coexisting phases. Blencoe discusses the role of crystal structure, and associated problems, in the calculation of solution parameters. 

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