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The kinetic electron trapping process in a shallow defect state and its subsequent thermal- or photo-stimulated promotion to a conduction band, followed by recombination in another defect, was described by Adirovitch using coupled rate differential equations. The solution for these equations has been frequently computed using the Runge-Kutta method. In this research, we empirically demonstrated that using the Runge-Kutta Fourth Order method may lead to incorrect and ramified results if the numbers of steps to achieve the solutions is not “large enough”. Taking into account these results, we conducted numerical analysis and experiments to develop an algorithm that determines the smallest non-critical number of steps in an automatic way to optimize the application of the Runge-Kutta Fourth Order method. This algorithm was implemented and tested in a variety of situations and the results have shown that our solution is robust in dealing with different equations and parameters.

Adirovitch proposed a description of the luminescence in crystal phosphor through a kinetic process involving electrons trapped in a crystal defect, their subsequent thermally or optically induced promotion to a conduction band, and their recombination in another defect [

In Adirovitch’s model, the concentration of trapped charges () and free charges in the conduction band () are related by the rate equations:

where, and are positive adjustable parameters associated with the release of charges from traps, retrapping and the charge and electron-hole recombinetion process, respectively. In general the energy dissipation of the recombination process occurs through a radiation process. Thus, the function presented in Equation (1) is proportional to the luminescence in crystal phosphor and is the concentration of traps [

Previous research findings have shown that the solutions for the non-linear stability analysis of equations 1a and 1b are stable, following a hyperbolic path [

In this paper we initially report numerical analyses of the fourth order Runge-Kutta method as applied to the solution of Adirovitch model Equations (1a) and (1b). Even though the RK method is stable, we identified a disconcerting property that emerges from the stiffness of the method when solving these equations: the numerical results are reliable and efficient only for a limited range of parameter values. To solve this problem, this paper proposes an algorithm to determine the best values for parameters in the Runge-Kutta method in order to guarantee the reliability of the results while using the smallest numbers of steps to reach a solution. This algorithm was implemented and has been used in different situations with different parameters. Thus far, to the best of our knowledge, the proposed algorithm has not produced any incorrect solutions.

The paper is organized as follows: in the first section we present the numerical analysis of the Adirovitch model; next, we present the problem of calculating the smallest number of steps to find the solution of this model using a Runge-Kutta Method; and finally we present an algorithm, together with its implementation and tests, to optimize the calculation of the smallest noncritical number of steps to reach the solution.

To numerically solve the Equations (1a) and (1b) for the Adirovitch model presented in the introduction of this work it is necessary to find the correct values for each adjustable parameter, , and. Powerful genetic [

Equations (1) are made more tractable by introducing the transformations:

where is the total concentration of trapped charges at the initial instant, and. Since the parameters, and enter linearly in the right hand side of Equations (1), mathematically only two of them are really independent. Therefore, it is natural to make an additional scaling. Then, using , we deal with only two parameters. Thus, Equations (1) are written as:

Numerical methods for the solution of Equations (3) can be divided into two categories [

The numerical solution of Equations (3) belonging to the first category are mostly grouped as Runge-Kutta [21, 22,27-30], Bulirsch-Stoer [

A widely used form of the Runge-Kutta method is of the fourth order. Using a vector notation typical in differential equations, and , the advancing formula from time to is given by:

where is the time increment for N steps in which the interval is divided,

The precision of is proportional to. For orders greater than four, it is necessary to evaluate M to M + 2 more functions, increasing the computational cost. The equilibrium between computational efficiency and cost has been found for the fourth order Runge-Kutta method [13,23].

As the error is proportional to, the Runge-Kutta method has a strong dependence on the number of steps. It predicts that the solution is invariant for large numbers of steps and that errors accumulate with any decrease in the number of steps.

Because of the dynamic nature of the Equations (3), it is possible that the accumulation of error makes the solutions erratic. Therefore, we investigated the types of errors that can happen depending on the number of steps and the values of the parameters of the Equations (3).

We have found at least four types of erratic solutions for Equations (3), which are shown in

The existence of incorrect solutions can be evaluated by analysing the correction for the n-th step given by.

In

of the result of the sum of these differences. In fact we verified through an exact numerical solution of Equations (3) that wrong solutions for N = 170, 200, 230 and 270 begin at approximately, respectively. Therefore, is not capable of revealing cumulative errors and a new algorithm must be considered in order to do so.

The analysis given in Section 2 reveals that the Equations (3) behave chaotically depending on the number of steps. Previous research findings have tried to provide mathematical explanations for why such phenomena occur in nonlinear systems [

The Runge-Kutta method numerically solves Equations (3) if the solution in a given interval becomes independent of the number of steps (i.e. the number of steps is large enough). However, as discussed in Section 2, we find that can detect fin-like, doubling and divergence (

In Section 3, we observed that incorrect solutions may occur from the beginning of the calculations. For example, considering the same interval of integration, for, , which gives wrong results, in the first step is 0.954288... and for , , which gives correct results, it is 0.952203... after 10 steps (the interval is the same in both calculations). This result shows that there is a difference from the beginning of the calculation. On the other hand for N = 2000, p_{1} = 1, p_{2} = p_{3} = 65, the result of the calculation in the first step is 0.995014870675078 and for N = 20,000 after 10 steps the same level of numerical precision is achieved.

By examining these results, we conclude that the result of the application of the RK method using the interval of the first step and dividing it into 10 steps may reveal whether the number of steps will lead to incorrect results. So we will build the algorithm considering only the interval of the first step. We calculate the result of the application of the RK method with a single step in this interval and subtract the result of the calculation obtained by dividing this same interval into M steps. We write this algorithm as:

Let us consider the difference where and are calculated with one and 10 steps in the interval, respectively. Then, since we are dealing with an expansion of the fourth order, we expect that to produce a correct solution this difference must be smaller than. In case of incorrect solutions, the difference does not maintain proportional to and will increase.

_{2} = 65 and. The features of and are very similar, until the second arrow; above the second arrow there are differences in the structure of the results. Therefore, the best description of the error in the application of the Runge-Kutta method in the solution of Equations (3) may include both and features.

To include the behavior of and, as suggested by the analysis of _{1} and y_{2}. Therefore we write the new algorithm as:

, with and (a), (b), (c), and (d).

() shows the exact solution region until.

The results of and are similar for small values of the number of steps (and). However when the number of steps increases closer to the number ideal of steps, we observed that reveals more details. For this reason, we adopted the algorithm to determine the smallest number of steps with which it is possible to reliably solve Equations (3) using the RK method.