Compact schemes in time with applications to partial differential equations

We propose a new class of fourth-and sixth-order schemes in time for parabolic and hyperbolic equations. The method follows the compact scheme methodology by elaborating implicit relations between the approximations of the function and its derivatives. We produce a series of A-stable methods with low dispersion and high accuracy. Several benchmarks for linear and non-linear Ordinary Differential Equations demonstrate the effectiveness of the method. Then a second set of numerical benchmarks for Partial Differential Equations such as convection-diffusion, Schrodinger equation, wave equation, Burgers, and Euler system give the numerical evidences of the superior advantage of the method with respect to the traditional Runge-Kutta or multistep methods.S. Clain and G.J. Machado acknowledge the financial support by Portuguese Funds through Foundation for Science and Technology (FCT) in the framework of the Strategic Funding UIDB/04650/2020. M.T. Malheiro acknowledges the financial support by Portuguese Funds through Foundation for Science and Technology (FCT) in the framework of the Projects UIDB/00013/2020 and UIDP/00013/2020 of CMAT-UM. S. Clain, G.J. Machado, and M.T. Malheiro acknowledge the fi-nancial support by FEDER - Fundo Europeu de Desenvolvimento Regional, through COMPETE 2020 - Programa Operacional Fatores de Competitividade, POCI-01-0145-FEDER-028118 and PTDC/MAT-APL/28118/2017

A new compact finite difference scheme for solving the complex Ginzburg-Landau equation

a b s t r a c t The complex Ginzburg-Landau equation is often encountered in physics and engineering applications, such as nonlinear transmission lines, solitons, and superconductivity. However, it remains a challenge to develop simple, stable and accurate finite difference schemes for solving the equation because of the nonlinear term. Most of the existing schemes are obtained based on the Crank-Nicolson method, which is fully implicit and must be solved iteratively for each time step. In this article, we present a fourth-order accurate iterative scheme, which leads to a tri-diagonal linear system in 1D cases. We prove that the present scheme is unconditionally stable. The scheme is then extended to 2D cases. Numerical errors and convergence rates of the solutions are tested by several examples

Integration of the hyperbolic telegraph equation in (1+1) dimensions via the generalized differential quadrature method

AbstractThe 2D generalized differential quadrature method (hereafter called ((1+1)-GDQ) is introduced within the context of dynamical system for solving the hyperbolic telegraph equation in (1+1) dimensions. Best efficiency is obtained with a low-degree polynomial (n⩽8) for both time variable and x-direction. From realistic examples, some models are presented to illustrate an excellent performance of the proposed method, compared with the exact results

Exponential integrators: tensor structured problems and applications

The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as phi-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed

Spatio-temporal integral equation methods with applications

Electromagnetic interactions are vital in many applications including physics, chemistry, material sciences and so on. Thus, a central problem in physical modeling is the electromagnetic analysis of materials. Here, we consider the numerical solution of the Maxwell equation for the evolution of the electromagnetic field given the charges, and the Newton or Schr\\"odinger equation for the evolution of particles. By combining integral equation techniques with new spectral deferred correction algorithms in time and hierarchical methods in space, we develop fast solvers for the calculation of electromagnetism with relaxations of the model in different scenarios. The dissertation consists of two parts, aiming to resolve the challenges in the temporal and spatial direction, respectively. In the first part, we study a new class of time stepping methods for time-dependent differential equations. The core algorithm uses the pseudo-spectral collocation formulation to discretize the Picard type integral equation reformulation, producing a highly accurate and stable representation, which is then solved via the deferred correction technique. By exploiting the mathematical properties of the formulation and the convergence procedure, we develop some new preconditioning techniques from different perspectives that are accurate, robust, and can be much more efficient than existing methods. As is typical of spectral methods, the solution to the discretization is spectral accurate and the time step-size is optimal, though the cost of solving the system can be high. Thus, the solver is particularly suited to problems where very accurate solutions are sought or large time-step is required, e.g., chaotic systems or long-time simulation. In the second part, we study the hierarchical methods with emphasis on the spatial integral equations. In the first application, we implement a parallel version of the adaptive recursive solver for two-point boundary value problem by Cilk multithreaded runtime system based on the integral equation formulation. In the second application, we apply the hierarchical method to two-layered media Helmholtz equations in the acoustic and electromagnetic scattering problems. With the method of images and integral representations, the spatially heterogeneous translation operators are derived with rigorous error analysis, and the information is then compressed and spread in a fashion similar to fast multipole methods. The preliminary results suggest that our approach can be faster than existing algorithms with several orders of magnitude. We demonstrate our solver on a number of examples and discuss various useful extensions. Preliminary results are favorable and show the viability of our techniques for integral equations. Such integral equation methods could well have a broad impact on many areas of computational science and engineering. We describe further applications in biology, chemistry, and physics, and outline some directions for future work.Doctor of Philosoph

Time-dependent coupled-cluster for ultrafast spectroscopy

The ultimate reason for chemical reactivity is the electronic motion, occurring at an attosecond timescale. Until the last century, it was impossible to observe it directly, as the shortest available laser pulses had duration in the order of femtoseconds. Recent technological advances lead to sub-femtosecond laser pulses, making possible real-time observation and control of electron dynamics.My Ph.D. thesis aims to develop and implement a model for the interaction between ultrashort laser pulses and molecules. This is interesting as an extension of the theory and the computational tools available, to design experiments at laser facilities, and to predict and interpret their outcomes.The theoretical framework that we have chosen is the time-dependent coupled-cluster (TDCC) theory. We have implemented our code in the eT program, which represents the first released implementation of a TDCC method.After validating our procedures by comparison with the literature, we used our code to calculate the electronic response to a pump-probe sequence of laser pulses. We performed convergence tests of parameters on the LiH. Then, we observed and interpreted the effect of the delay between pump and probe pulses on the LiF transient absorption spectrum.We extended this implementation to a time-dependent equation-of-motion coupled-cluster (TD-EOM-CC) approach with the use of a reduced basis calculated with an asymmetric band Lanczos algorithm, and within the core-valence separation (CVS) approximation. This converged to the same spectral features as the TDCC but with much lower computational times, as we showed for LiF. We observed the limits of CVS approximation: for the LiH molecule, several peaks were not correctly retrieved. Finally, we modeled the transient absorption for the glycine molecule, which is a good candidate for experimental investigations.We also modeled the electronic impulsive stimulated Raman scattering (ISXRS) population transfer induced by an ultrashort laser pulse through the TD-EOM-CC model for Ne, CO, pyrrole, and p-aminophenol and visualized through a movie the real-time evolution of the electronic density of p-aminophenol.The significance of this work lies in the development of theoretical and computational tools to be used in attochemistry: one groundbreaking application can be the direct control of electrons, which would have a big impact on many research fields, like medicine, biology, and material science

Fingerprints in the Optical and Transport Properties of Quantum Dots

The book "Fingerprints in the optical and transport properties of quantum dots" provides novel and efficient methods for the calculation and investigating of the optical and transport properties of quantum dot systems. This book is divided into two sections. In section 1 includes ten chapters where novel optical properties are discussed. In section 2 involve eight chapters that investigate and model the most important effects of transport and electronics properties of quantum dot systems This is a collaborative book sharing and providing fundamental research such as the one conducted in Physics, Chemistry, Material Science, with a base text that could serve as a reference in research by presenting up-to-date research work on the field of quantum dot systems