A system of ordinary differential equations can represent many phenomena of interest in physiology and biochemistry. Such a model describes reactions and diffusion among several chemical species. The number of these species in real problems can be very large, so this fact must be taken into consideration in each numerical method for finding a good approximate solution. Very well known explicit schemes such as the Euler or the Runge-Kutta method are not very convenient for such a problem due to their conditional stability, that means, the time step must be very small. This is fatal, especially, for stiff problems, and, that is why, many studies and new methods for stiff problems have been developed in the past few decades.
In the present paper, two new absolutely stable explicit schemes applicable to a linear reaction system are presented. First, the two types of reaction systems are presented – reversible reaction of two species, namely their concentrations, and circular reactions of three species. For this simple case, the analytical solution is presented in case of reversible reactions. This can be done for circular reactions too, but, with increasing number of species, it is a quite difficult task. For the circular reaction case, two new numerical methods are presented.
The basic idea for both methods consists in splitting a circular reaction into a chain of reversible reactions for which one can find the exact solution. The second algorithm improves the first algorithm in such a way that the result is an average of two results. Algorithm 1 is computed twice and for the second time the reverse order of spliting is used.
The absolute stability for both algorithms is proved. Moreover, for the first algorithm, a first order of convergence, and, for the second algorithm, a second order of convergence is proved. These theoretical results are numerically confirmed on experiments where the order of convergence for both methods is computed and the accuracy of both methods is compared with the one of Gear’s methods, a two step backward differentiation formula (BDF2) which is an A-stable method of order 2.