An efficient and fast parallel method for Volterra integral equations of Abel type

In this paper we present an efficient and fast parallel waveform relaxation method for Volterra integral equations of Abel type, obtained by reformulating a nonstationary waveform relaxation method for systems of equations with linear coefficient constant kernel. To this aim we consider the Laplace transform of the equation and here we apply the recurrence relation given by the Chebyshev polynomial acceleration for algebraic linear systems. Back in the time domain, we obtain a three term recursion which requires, at each iteration, the evaluation of convolution integrals, where only the Laplace transform of the kernel is known. For this calculation we can use a fast convolution algorithm. Numerical experiments have been done also on problems where it is not possible to use the original nonstationary method, obtaining good results in terms of improvement of the rate of convergence with respect the stationary method

Multistep collocation methods for Volterra Integral Equations

We introduce multistep collocation methods for the numerical integration of Volterra Integral Equations, which depend on the numerical solution in a fixed number of previous time steps. We describe the constructive technique, analyze the order of the resulting methods and their linear stability properties. Numerical experiments confirm the theoretical expectations

Two classes of linearly implicit numerical methods for stiff problems: analysis and MATLAB software

The purpose of this work lies in the writing of efficient and optimized Matlab codes to implement two classes of promising linearly implicit numerical schemes that can be used to accurately and stably solve stiff Ordinary Differential Equations (ODEs), and also Partial Differential Equations (PDEs) through the Method Of Lines (MOL). Such classes of methods are the Runge-Kutta (RK) [28] and the Peer [17], and have been constructed using a variant of the Exponential-Fitting (EF) technique [27]. We carry out numerical tests to compare the two methods with each other, and also with the well known and very used Gaussian RK method, by the point of view of stability, accuracy and computational cost, in order to show their convenience

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

Non-stationary wave relaxation methods for general linear systems of Volterra equations: convergence and parallel GPU implementation

Producción CientíficaIn the present paper, a parallel-in-time discretization of linear systems of Volterra equations of type u¯(t)=u¯0+∫t0K(t−s)u¯(s) d s+f¯(t),0<t≤T, is addressed. Related to the analytical solution, a general enough functional setting is firstly stated. Related to the numerical solution, a parallel numerical scheme based on the Non-Stationary Wave Relaxation (NSWR) method for the time discretization is proposed, and its convergence is studied as well. A CUDA parallel implementation of the method is carried out in order to exploit Graphics Processing Units (GPUs), which are nowadays widely employed for reducing the computational time of several general purpose applications. The performance of these methods is compared to some sequential implementation. It is revealed throughout several experiments of special interest in practical applications the good performance of the parallel approach.Ministerio de Universidades e Investigación de Italia (MUR), a través del proyecto PRIN 2017 (No. 2017JYCLSF) “Aproximación preservadora de estructuras de problemas evolutivos”Publicación en abierto financiada por el Consorcio de Bibliotecas Universitarias de Castilla y León (BUCLE), con cargo al Programa Operativo 2014ES16RFOP009 FEDER 2014-2020 DE CASTILLA Y LEÓN, Actuación:20007-CL - Apoyo Consorcio BUCL

High performance parallel numerical methods for Volterra equations with weakly singular kernels

Non-stationary discrete time waveform relaxation methods for Abel systems of Volterra integral equations using fractional linear multistep formulae are introduced. Fully parallel discrete waveform relaxation methods having an optimal convergence rate are constructed. A significant expression of the error is proved, which allows us to estimate the number of iterations needed to satisfy a prescribed tolerance and allows us to identify the problems where the optimal methods offer the best performance. The numerical experiments confirm the theoretical expectations

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

Some new uses of the η_m(Z) functions

We present a procedure and a MATHEMATICA code for the conversion of formulae expressed in terms of the trigonometric functions sin(omega x), cos(omega x) or hyperbolic functions sinh(lambda x), cosh(lambda x) to forms expressed in terms of eta(m)(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