Raffaele D’Ambrosio

Parameter estimation in exponentially fitted hybrid methods for second order differential problems

In this work we deal with exponentially fitted methods for the numerical solution of second order ordinary differential equations, whose solutions are known to show a prominent exponential behaviour depending on the value of an unknown parameter to be suitably determined. The knowledge of an estimation to the unknown parameter is needed in order to apply the numerical method, since its coefficients depend on the value of the parameter. We present a strategy for the practical estimation of the parameter, which is also tested on some selected problems.

Exponentially fitted two-step Runge–Kutta methods: Construction and parameter selection

Abstract We derive exponentially fitted two-step Runge–Kutta methods for the numerical solution of y ′ = f ( x , y ) , specially tuned to the behaviour of the solution. Such methods have nonconstant coefficients which depend on a parameter to be suitably estimated. The construction of the methods is shown and a strategy of parameter selection is presented. Some numerical experiments are provided to confirm the theoretical expectations.

Revised exponentially fitted Runge–Kutta–Nyström methods

Abstract It is the purpose of this paper to revise the exponential fitting technique for the numerical solution of special second order ordinary differential equations (ODEs) y ″ = f ( x , y ) , with oscillatory or periodic solutions, by Runge–Kutta–Nystrom methods. Due to the multistage nature of these methods, the proposed technique takes into account the contribution to the error arising from the computation of the internal stages. The benefit on the accuracy of the overall numerical scheme is visible in the presented numerical evidence.

Numerical solution of a diffusion problem by exponentially fitted finite difference methods

This paper is focused on the accurate and efficient solution of partial differential differential equations modelling a diffusion problem by means of exponentially fitted finite difference numerical methods. After constructing and analysing special purpose finite differences for the approximation of second order partial derivatives, we employed them in the numerical solution of a diffusion equation with mixed boundary conditions. Numerical experiments reveal that a special purpose integration, both in space and in time, is more accurate and efficient than that gained by employing a general purpose solver.

Adapted numerical methods for advection–reaction–diffusion problems generating periodic wavefronts

Abstract The paper presents an adapted numerical integration for advection–reaction–diffusion problems. The numerical scheme, exploiting the a-priori knowledge of the qualitative behaviour of the solution, gains advantages in terms of efficiency and accuracy with respect to classic schemes already known in literature. The adaptation is here carried out through the so-called trigonometrical fitting technique for the discretization in space, giving rise to a system of ODEs whose vector field contains both stiff and non-stiff terms. Due to this mixed nature of the vector field, an Implicit–Explicit (IMEX) method is here employed for the integration in time, based on the first order forward–backward Euler method. The coefficients of the method here introduced rely on unknown parameters which have to be properly estimated. In this work, such an estimate is performed by minimizing the leading term of the local truncation error. The effectiveness of this problem-oriented approach is shown through a rigorous theoretical analysis and some numerical experiments.

Runge‐Kutta‐Nyström Stability for a Class of General Linear Methods for y″ = f(x,y)

Aim of this paper is to begin an investigation on the linear stability analysis of a class of General Linear Methods for special second order Ordinary Differential Equations, taking as a starting point a class of two‐step Runge‐Kutta‐Nystrom methods, which is obtained through a transformation of the two‐step Runge‐Kutta methods. We construct methods with one and two stages possessing the same stability properties of the indirect collocation Gauss‐Legendre Runge‐Kutta‐Nystrom methods, which are P‐stable.

Advances on Collocation Based Numerical Methods for Ordinary Differential Equations and Volterra Integral Equations

We present a survey on collocation based methods for the numerical integration of Ordinary Differential Equations (ODEs) and Volterra Integral Equations (VIEs), starting from the classical collocation methods, to arrive to the most important modifications appeared in the literature, also considering the multistep case and the usage of basis of functions other than polynomials.

Stochastic Numerical Models of Oscillatory Phenomena

The use of time series for integrating ordinary differential equations to model oscillatory chemical phenomena has shown benefits in terms of accuracy and stability. In this work, we suggest to adapt also the model in order to improve the matching of the numerical solution with the time series of experimental data. The resulting model is a system of stochastic differential equations. The stochastic nature depends on physical considerations and the noise relies on an arbitrary function which is empirically chosen. The integration is carried out through stochastic methods which integrate the deterministic part by using one-step methods and approximate the stochastic term by employing Monte Carlo simulations. Some numerical experiments will be provided to show the effectiveness of this approach.

Multi-value Numerical Methods for Hamiltonian Systems

We discuss the effectiveness of multi-value numerical methods in the numerical treatment of Hamiltonian problems. Multi-value (or general linear) methods extend the well-known families of Runge-Kutta and linear multistep methods and can be considered as a general framework for the numerical solution of ordinary differential equations. There are some features that needs to be achieved by reliable geometric numerical integrators based on multi-value methods: G-symplecticity, symmetry and boundedness of the parasitic components. In particular, we analyze the effects of the mentioned features for the long term conservation of the energy and provide the numerical evidence confirming the theoretical expectations.

Adapted numerical modelling of the Belousov–Zhabotinsky reaction

Adapted numerical schemes for the integration of differential equations generating periodic wavefronts have reported benefits in terms of accuracy and stability. This work is focused on differential equations modelling chemical phenomena which are characterized by an oscillatory dynamics. The adaptation is carried out through the exponential fitting technique, which is specially suitable to follow the apriori known qualitative behavior of the solution. In particular, we have merged this strategy with the information coming from existing theoretical studies and especially the observation of time series. Numerical tests will be provided to show the effectiveness of this problem-oriented approach.