Eleonora Messina

Parallel Adams methods

Abstract In the literature, various types of parallel methods for integrating nonstiff initial-value problems for first-order ordinary differential equation have been proposed. The greater part of them are based on an implicit multistage method in which the implicit relations are solved by the predictor-corrector (or fixed point iteration) method. In the predictor-corrector approach the computation of the components of the stage vector iterate can be distributed over s processors, where s is the number of implicit stages of the corrector method. However, the fact that after each iteration the processors have to exchange their just computed results is often mentioned as a drawback, because it implies frequent communication between the processors. Particularly on distributed memory computers, such a fine grain parallelism is not attractive. An alternative approach is based on implicit multistage methods which are such that the implicit stages are already parallel, so that they can be solved independently of each other. This means that only after completion of a step, the processors need to exchange their results. The purpose of this paper is the design of a class of parallel methods for solving nonstiff IVPs. We shall construct explicit methods of order k + 1 with k parallel stages where each stage equation is of Adams-Bashforth type and implicit methods of order k + 2 with k parallel stages which are of Adams-Moulton type. The abscissae in both families of methods are proved to be the Lobatto points, so that the Adams-Bashforth type method can be used as a predictor for the Adams-Moulton-type corrector.

Stability analysis of delayed SIR epidemic models with a class of nonlinear incidence rates

We analyze stability of equilibria for a delayed SIR epidemic model, in which population growth is subject to logistic growth in absence of disease, with a nonlinear incidence rate satisfying suitable monotonicity conditions. The model admits a unique endemic equilibrium if and only if the basic reproduction number R0 exceeds one, while the trivial equilibrium and the disease-free equilibrium always exist. First we show that the disease-free equilibrium is globally asymptotically stable if and only if R0 1. Second we show that the model is permanent if and only if R0 > 1. Moreover, using a threshold parameter R0 characterized by the nonlinear incidence function, we establish that the endemic equilibrium is locally asymptotically stable for 1 < R0 R0 and it loses stability as the length of the delay increases past a critical value for 1 < R0 < R0. Our result is an extension of the stability results in (J-J. Wang, J-Z. Zhang, Z. Jin, Analysis of an SIR model with bilinear incidence rate, Nonl. Anal. RWA. 11 (2009) 2390-2402).

General linear methods for Volterra integral equations

We investigate the class of general linear methods of order p and stage order q=p for the numerical solution of Volterra integral equations of the second kind. Construction of highly stable methods based on the Schur criterion is described and examples of methods of order one and two which have good stability properties with respect to the basic test equation and the convolution one are given.

Convergence of solutions for two delays Volterra integral equations in the critical case

Abstract In this paper, for the “critical case” with two delays, we establish two relations between any two solutions y ( t ) and y ∗ ( t ) for the Volterra integral equation of non-convolution type y ( t ) = f ( t ) + ∫ t − τ t − δ k ( t , s ) g ( y ( s ) ) d s and a solution z ( t ) of the first order differential equation z ( t ) = β ( t ) [ z ( t − δ ) − z ( t − τ ) ] , and offer a sufficient condition that lim t → + ∞ ( y ( t ) − y ∗ ( t ) ) = 0 .

Comparing analytical and numerical solution of a nonlinear two-delay integral equations

Abstract: Numerical solution of two delays Volterra Integral Equations is considered and the stability is studied on a nonlinear test equation by carrying out a parallel investigation both on the continuous and the discrete problem.

Parallel linear system solvers for Runge-Kutta-Nyström methods

Abstract Solving the nonlinear systems arising in implicit Runge-Kutta-Nystrom type methods by (modified) Newton interation leads to linear systems whose matrix of coefficients is of the form I − A ⊗ h2J where A is the Runge-Kutta-Nystrom matrix and J an approximation to the Jacobian of the right-hand-side function of the system of differential equations. For larger systems of differential equations, the solution of these linear systems by a direct linear solver is very costly, mainly because of the LU-decomposition. We try to reduce these costs by solving the linear Newton systems by an inner iteration process. Each inner iteration again requires the solution of a linear system. However, the matrix of coefficients in these new linear systems are of the form I − B ⊗ h2J where B is a nondefective matrix with positive eigenvalues, so that by a similarity transformation, we can decouple the system into subsystems the dimension of which equals the dimension of the system of differential equations. Since the subsystems can be solved in parallel, the resulting integration method is highly efficient on parallel computer systems. The performance of the parallel iterative linear system method for Runge-Kutta-Nystrom equations (PILSRKN method) is illustrated by means of a few examples from the literature.

Convergence of a numerical method for the solution of non-standard integro-differential boundary value problems

In a recent paper we proposed a numerical method to solve a non-standard non-linear second order integro-differential boundary value problem. Here, we answer two questions remained open: we state the order of convergence of this method and provide some sufficient conditions for the uniqueness of the solution both of the discrete and the continuous problem. Finally, we compare the performances of the method for different choices of the iteration procedure to solve the non-standard nonlinearity.

Stability analysis of linear Volterra equations on time scales under bounded perturbations

Abstract We analyze the stability of the zero solution to Volterra equations on time scales with respect to two classes of bounded perturbations. We obtain sufficient conditions on the kernel which include some known results for continuous and for discrete equations. In order to check the applicability of these conditions, we apply the theory to a test example.

Nonlinear stability of direct quadrature methods for Volterra integral equations

An important topic in the numerical analysis of Volterra integral equations is the stability theory. The main results known in the literature have been obtained on linear test equations or, at least, on nonlinear equations with convolution kernel. Here, we consider Volterra integral equations with Hammerstein nonlinearity, not necessarily of convolution type, and we study the error equation for Direct Quadrature methods with respect to bounded perturbations. For a class of Direct Quadrature methods, we obtain conditions on the stepsize h for the numerical solution to behave stably and we report numerical examples which show the robustness of this nonlinear stability theory.

A sufficient condition for the stability of direct quadrature methods for Volterra integral equations

Within the theoretical framework of the numerical stability analysis for the Volterra integral equations, we consider a new class of test problems and we study the long-time behavior of the numerical solution obtained by direct quadrature methods as a function of the stepsize. Furthermore, we analyze how the numerical solution responds to certain perturbations in the kernel.