Resolution of Initial Value Problems of Ordinary Differential Equations Systems

In this work, we present some techniques applicable to Initial Value Problems when solving a System of Ordinary Differential Equations (ODE). Such techniques should be used when applying adaptive step-size numerical methods. In our case, a Runge-Kutta-Fehlberg algorithm (RKF45) has been employed, but the procedure presented here can also be applied to other adaptive methods, such as N-body problems, as AP3M or similar ones. By doing so, catastrophic cancellations were eliminated. A mathematical optimization was carried out by introducing the objective function in the ODE System (ODES). Resizing of local errors was also utilised in order to adress the problem. This resize implies the use of certain variables to adjust the integration step while the other variables are used as parameters to determine the coefficients of the ODE system. This resize was executed by using the asymptotic solution of this system. The change of variables is necessary to guarantee the stability of the integration. Therefore, the linearization of the ODES is possible and can be used as a powerful control test. All these tools are applied to a physical problem. The example we present here is the effective numerical resolution of Lemaitre-Tolman-Bondi space-time solutions of Einstein Equations.