Resolution of Initial Value Problems of Ordinary Differential Equations Systems

Josep Vicent Arnau I Córdoba,Màrius Josep Fullana I Alfonso

doi:10.3390/math10040593

Josep Vicent Arnau I Córdoba, Màrius Josep Fullana I Alfonso

Open Access

https://doi.org/10.3390/math10040593

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Abstract

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.

Highlights

In mathematics, a dynamical system is a system in which a function describes the time-dependence of a point in a geometrical space
Some important definitions related with our problem are presented in Examples of modern physical problems where analytic solutions of differential equations are not possible, such as the problem we present in the present paper, is found in quantum systems
We enter the phase of incorporating our objective function into the dynamical system that we have modelled through its analytical derivation so that it can evolve with the global model

Summary

Introduction

A dynamical system is a system in which a function describes the time-dependence of a point in a geometrical space. In order to make a prediction about the system’s future behaviour, an analytical solution of such equations or their integration over time through computer simulations must be carried out. When modelling such a dynamical system to be able to investigate certain aspects of its behaviour, we end up generating a system of ordinary differential equations (ODES), partial differential equations (PDE), and algebraic equations (AE), whose analogous solution is unknown to us in the vast majority of cases. We need to resort to numerical methods for ODES which allow us to evaluate approximate solutions with a certain tolerance after fixing the indeterminate constants of the system, either by initial values (IVP or initial value problem) or boundary conditions (boundary Cauchy problem). We will find a large number of numerical methods at our disposal that can be divided into two large groups:

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