Implicit Difference Approximation Research Articles

Theory of Cyclic Voltammetry at Microdisk Electrodes

AbstractThe cyclic voltammetric response, which is obtained by measurements at microdisk electrodes (R ± 1 mm), may deviate strongly from theoretical predictions. The cause of such effects is due to nonlinear diffusion processes, which, in principle, are present at unshielded disk electrodes. They increase with diminishing electrode radii and slow scan rates in accordance with the value of the dimensionless parameter = (D/aR2)1/2. A quantitative description of nonlinear mass transport to microdisk electrodes can be achieved with the aid of a two‐dimensional simulation based on an implicit difference approximation. The agreement with experimental data is very close. The reversible and the quasi‐reversible charge transfer is discussed. To facilitate the interpretation of cyclic voltammograms, working‐curves and diagnostic criteria are presented.

Diffusion progresses at finite (micro) disk electrodes solved by digital simulation

A modified simulation model based on a two-dimensional implicit difference approximation (ADI method) has been developed to describe diffusion processes at stationary unshielded (micro) disk electrodes. Besides the semi-infinite linear component, the model also takes into account a radial diffusion gradient which is always present in all transport processes of this nature. The current-time relationship for diffusion-controlled electrolysis was used to test the consistency and accuracy of the numerical approximation. The computation reveals, in close agreement with experimental data, that the familiar Cottrell equation should be replaced by the formula:

Diffusion processes at finite (micro) disk electrodes solved by digital simulation

A modified simulation model based on a two-dimensional implicit difference approximation (ADI method) has been developed to describe diffusion processes at stationary unshielded (micro) disk electrodes. Besides the semi-infinite linear component, the model also takes into account a radial diffusion gradient which is always present in all transport processes of this nature. The current-time relationship for diffusion-controlled electrolysis was used to test the consistency and accuracy of the numerical approximation. The computation reveals, in close agreement with experimental data, that the familiar Cottrell equation should be replaced by the formula:

Propagation of planar shock waves in an exponential atmosphere

We consider planar explosions in a medium with an exponential density distribution. In contrast to the so-called sectorial approximation [1] we take into account the overflow of energy from a lower region to an upper region, so that our solution of the problem considered here gives a truer picture of the flow of the gas at a later stage of a point explosion in a nonhomogeneous atmosphere. The numerical solution in both upper and lower regions of the flow merges into the corresponding limiting self-similar regime [2, 3]. The calculations are carried out up to a “gap” in the atmosphere [4]. The computational method is based on implicit difference approximations.

Mesh refinements for parabolic equations of second order

Given certain implicit difference approximations to u t = a ( x ) u x x + b ( x ) u x + c ( x ) u {u_t} = a(x){u_{xx}} + b(x){u_x} + c(x)u in the region − ∞ > x > ∞ , t ≧ 0 - \infty > x > \infty ,t \geqq 0 , with a finer x-mesh width in the left half-plane than in the right, we consider the stability in the maximum norm of these schemes using several different interface conditions (at x = 0 x = 0 ). In order to obtain our results, we first prove a stability theorem for certain simple second-order parabolic initial boundary systems on an evenly spaced mesh in the right half-plane alone. By a standard procedure, the first problem is converted into the second one, and solved in this manner.

On the Accuracy of Implicit Difference Approximations to the Equation of Heat Flow

On the Accuracy of Implicit Difference Approximations to the Equation of Heat Flow