Dajana Conte

Two-step almost collocation methods for Volterra integral equations

In this paper we construct a new class of continuous methods for Volterra integral equations. These methods are obtained by using a collocation technique and by relaxing some of the collocation conditions in order to obtain good stability properties.

Two-step Runge-Kutta Methods with Quadratic Stability Functions

We describe the construction of implicit two-step Runge-Kutta methods with stability properties determined by quadratic stability functions. We will aim for methods which are A-stable and L-stable and such that the coefficients matrix has a one point spectrum. Examples of methods of order up to eight are provided.

Numerical search for algebraically stable two-step almost collocation methods

We investigate algebraic stability of the new class of two-step almost collocation methods for ordinary differential equations. These continuous methods are obtained by relaxing some of the interpolation and collocation conditions to achieve strong stability properties together with uniform order of convergence on the whole interval of integration. We describe the search for algebraically stable methods using the criterion based on the Nyquist stability function proposed recently by Hill. This criterion leads to a minimization problem in one variable which is solved using the subroutine fminsearch from MATLAB. Examples of algebraically stable methods in this class are also presented.

A Family of Multistep Collocation Methods for Volterra Integral Equations

We introduce a family of multistep collocation methods for the numerical integration of Volterra Integral Equations, which depends on the numerical solution on a fixed number of previous time steps. We describe the constructive technique, discuss the order of the resulting methods, analyze their linear stability properties and give an example of one stage method.

A practical approach for the derivation of algebraically stable two-step Runge-Kutta methods

We describe an algorithm, based on a new strategy recently proposed by Hewitt and Hill in the context of general linear methods, for the construction of algebraically stable two-step Runge-Kutta methods. Using this algorithm we obtained a complete characterization of algebraically stable methods with one and two stages.

A family of Multistep Collocation Methods for Volterra Integro‐Differential Equations

In this paper we analyze a family of multistep collocation methods for Volterra Integro‐Differential Equations, with the aim of increasing the order of classical one‐step collocation methods without increasing the computational cost. We discuss the order of the constructed methods and present the stability analysis.

GPU-acceleration of waveform relaxation methods for large differential systems

It is the purpose of this paper to provide an acceleration of waveform relaxation (WR) methods for the numerical solution of large systems of ordinary differential equations. The introduced technique is based on the employ of graphics processing units (GPUs) in order to speed-up the numerical integration process. A CUDA solver based on WR-Picard, WR-Jacobi and red-black WR-Gauss-Seidel iterations is presented and some numerical experiments realized on a multi-GPU machine are provided.

Natural Volterra Runge-Kutta methods

A very general class of Runge-Kutta methods for Volterra integral equations of the second kind is analyzed. Order and stage order conditions are derived for methods of order p and stage order q = p up to the order four. We also investigate stability properties of these methods with respect to the basic and the convolution test equations. The systematic search for A- and V0-stable methods is described and examples of highly stable methods are presented up to the order p = 4 and stage order q = 4.

An exponentially fitted quadrature rule over unbounded intervals

A new class of quadrature formulae for the computation of integrals over unbounded intervals with oscillating integrand is illustrated. Such formulae are a generalization of the gaussian quadrature formulae by exploiting the Exponential Fitting theory. The coefficients depend on the frequency of oscillation, in order to improve the accuracy of the solution. The construction of the methods with 1, 2 and 3 nodes is described, together with the comparison of the order of accuracy with respect to classical formulae.

Some new uses of the ηm(Z) functions

We present a procedure and a MATHEMATICA code for the conversion of formulae expressed in terms of the trigonometric functions sin(ωx), cos(ωx) or hyperbolic functions sinh(λx), cosh(λx) to forms expressed in terms of ηm(Z) functions. The possibility of such a conversion is important in the evaluation of the coefficients of the approximation rules derived in the frame of the exponential fitting. The converted expressions allow, among others, a full elimination of the 0/0 undeterminacy, uniform accuracy in the computation of the coefficients, and an extended area of validity for the corresponding approximation formulae.