Non-standard finite difference methods in dynamical systems

Abstract

This thesis analyses numerical methods used in finding solutions of diferential equations. Numerical methods are viewed as discrete dynamical systems that give useful information on continuous dynamical systems defined by systems of (ordinary) diferential equations. We analyse non-standard finite difference schemes that have no spurious fixed-points compared to the dynamical system under consideration, the linear stability/instability property of the fixed-points being the same for both the discrete and continuous systems. We obtain a sharper condition for the elementary stability of the schemes. For more complex dynamical systems which are dissipative, we design schemes that replicate this property. Furthermore, we investigate the impact of the above analysis on the numerical solution of partial differential equations. We specifically focus on reaction-diffusion equations that arise in many fields of engineering and applied sciences. Often their solutions enjoy the follow- ing essential properties: Stability/instability of the fixed points for the space independent equation, the conservation of energy for the stationary equation, and boundedness and positivity. We design new non-standard finite diference schemes which replicate these properties. Our construction make use of three strategies: the renormalization of the denominator of the discrete derivative, non-local approximation of the nonlinear terms and simple functional relation between step sizes. Numerical results that support the theory are provided. Copyright