Explicit digital simulation

This is what the standard explicit digital simulation method does. Then, /(u+A ) is the slope at the point D, and if we go only half-way along the slope /(u(t)), to point E and then the rest of the way with the new slope at D, we end up at point F, which is much closer to point B, the true solution. This loose description is mathematically expressed as (assuming Eq. 5.2)... 

EXPLICIT DIGITAL SIMULATION OF THE DME, USING THE EXPANDING-PLANE... 

Many advances in digital simulation have taken place since the publication of the first edition. Some of these advances have occurred in hardware through the development of the personal computer. Others have taken place by the development of commercial software that will perform specific simulations or will create a computer environment (e.g., a spreadsheet) that will allow one to do simulations without having to write a computer program. Finally, there have been theoretical advances where newer implicit algorithms are used to solve the necessary partial differential equations more efficiently than is possible using the more intuitive explicit methods described herein. 

One of the main uses of digital simulation - for some workers, the only application - is for linear sweep (LSV) or cyclic voltammetry (CV). This is more demanding than simulation of step methods, for which the simulation usually spans one observation time unit, whereas in LSV or CV, the characteristic time r used to normalise time with is the time taken to sweep through one dimensionless potential unit (see Sect. 2.4.3) and typically, a sweep traverses around 24 of these units and a cyclic voltammogram twice that many. Thus, the explicit method is not very suitable, requiring rather many steps per unit, but will serve as a simple introduction. Also, the groundwork for the handling of boundary conditions for multispecies simulations is laid here. 

In the past, Matsue et al. have discussed this issue in a study related to the characterization of diaphorase-pattemed surfaces by SECM. Using a digital simulation based on the explicit finite difference method that considered the heterogenous enzyme reaction at the substrate, they generated steady-state current vs. distance profiles that depended on the surface concentration of the enzyme. Using these working curves, they quantified the surface concentration of the active immobilized diaphorase (134). Using, the electrochemical... 

Today, there is hardly a paper written in electrochemistry, that does not casually mention the use, in some way, of a computer (or several) or hardly an electrochemist not routinely using one (or more). This was not the case in 1964, when Feldberg and Auerbach published their first paper on digital simulation. At that time, computers were much slower, more expensive to use and the electrochemists using them were a minority. This partly explains the anomaly of the box method the explicit finite difference method had been known since at least 1928 (Courant et al) the Crank-Nicolson improvement since 1947 (and it was immediately used by an electrochemist (Randles, 1948)) but Feldberg developed his particular style independently. 

Engelman EE, Evans DH (1992) Explicit finite-difference digital simulation of the effects of rate-controlled product adsorption or deposition in double-potential-step chronocoulometry. J Electroanal Chem 331 739-749... 

Britz D, 0sterby O, Stmtwolf J, Svennesen TK (2002) High-order spatial discretisations in electrochemical digital simulations. 3. Combination with the explicit Runge-Kutta algorithm. Comput Chem 26 97-103... 

Marques da SUva B, Avaca LA, Gonzalez ER (1989) New explicit finite difference methods in the digital simulation of electrochemical problems. J Electroanal Chem 269 1-14... 

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