Numerical treatment of Evolutionary Oscillatory Problems

The purpose of this thesis is the construction, the analysis and the implementation of new efficient and accurate numerical methods for the solution of evolutionary oscillatory problems. More specifically the thesis concerns: the computation of oscillatory integrals over unbounded intervals, the solution of Volterra integral equations with periodic solution and of special second-order ordinary differential equations with periodic or exponentially decaying solution. Some applications, analytical properties and theoretical results are reported in Chapter 1. General purpose methods require a small stepsize in order to follow the oscillations of the solution, therefore they result inefficient. Our idea is to exploit the qualitative knowledge of the solution in order to construct new efficient and accurate numerical methods specially tuned on the problem. These methods are derived by using the Exponential Fitting (EF) theory. A more detailed description of this theory is provided in Chapter 2. In Chapter 3 a new class of exponentially fitted (ef) quadrature formulae for the computation of oscillatory infinite integrals is constructed and analyzed. The weights and the nodes of these ef formulae depend on the frequency of the integrand function and are solution of a nonlinear system. To solve this system we develop a suitable efficient version of Newton method. We study the error behaviour of these formulae and we prove that the error decreases as the oscillation increases. We build exponentially fitted Gauss-Laguerre rules with 1 up to 6 nodes. Numerical experiments on significant test examples confirm theoretical expectations and show that the ef Gauss-Laguerre rules are more efficient than the classical ones. In Chapter 4 an ef Direct Quadrature (DQ) method for the numerical solution of Volterra integral equations (VIEs) with periodic solution is described. This method is based on an ef two nodes Gaussian quadrature rule. The weights and the nodes of this quadrature formula depend on the frequency of the problem and are solution of a nonlinear system. This ef Gaussian DQ method requires the knowledge of the numerical solution in points not belonging to the mesh. Therefore we develop a suitable ef interpolation formula which depends on the frequency, too, and preserves the accuracy order of the whole method. The convergence analysis shows that the order of the ef DQ method is four like the classical Gaussian DQ method, but the error is smaller when periodic problems are treated. Various numerical experiments show that on test VIEs with oscillatory or periodic solution, at the same computational cost, the ef Gaussian DQ method is more accurate than the classical Gaussian DQ method even for approximated values of the frequency. In Chapter 5, we introduce multistage methods for special second order differential equations (ODEs) with periodic or with exponentially decaying solution. We introduce these multistage methods by considering the contributions of the stage errors in the overall numerical scheme and by means the EF technique. In this way we develop a revised version of some EF-based Runge-Kutta-Nyström (RKN) methods. The coefficients of these methods depend on the problem parameters, i.e. the frequency of oscillations, for periodic problems, or the exponent, for problems with exponentially decaying solution. Therefore we propose a suitable strategy to estimate these parameters when they are not available. This strategy is based on the annihilation of the principal term of the local truncation error and does not require further function evaluations. Then an example of revised explicit two-stages ef RKN method is given and some numerical experiments are carried out. These tests show that this ef-RKN method outperforms the standard one and that the parameter estimation is sharp. Finally, in Chapter 6 we propose a parallel quadrature algorithm on Graphics Processing Units (GPU) based on Newton-Cotes formulas. This is a preliminary step for the numerical solution of multidimensional integrals and discretized Volterra integral equations. We report some numerical results obtained by the multi-GPU cluster E4 belonging to the Department of Mathematics, University of Salerno