Numerical investigation on nonlocal problems with the fractional Laplacian

Nonlocal models have recently become a powerful tool for studying complex systems with long-range interactions or memory effects, which cannot be described properly by the traditional differential equations. So far, different nonlocal (or fractional differential) models have been proposed, among which models with the fractional Laplacian have been well applied. The fractional Laplacian (-Δ)α/2 represents the infinitesimal generator of a symmetric α-stable Lévy process. It has been used to describe anomalous diffusion, turbulent flows, stochastic dynamics, finance, and many other phenomena. However, the nonlocality of the fractional Laplacian introduces considerable challenges in its mathematical modeling, numerical simulations, and mathematical analysis. To advance the understanding of the fractional Laplacian, two novel and accurate finite difference methods -- the weighted trapezoidal method and the weighted linear interpolation method are developed for discretizing the fractional Laplacian. Numerical analysis is provided for the error estimates, and fast algorithms are developed for their efficient implementation. Compared to the current state-of-the-art methods, these two methods have higher accuracy but less computational complexity. As an application, the solution behaviors of the fractional Schördinger equation are investigated to understand the nonlocal effects of the fractional Laplacian. First, the eigenvalues and eigenfunctions of the fractional Schrödinger equation in an infinite potential well are studied, and the results provide insights into an open problem in the fractional quantum mechanics. Second, three Fourier spectral methods are developed and compared in solving the fractional nonlinear Schördinger equation (NLS), among which the SSFS method is more effective in the study of the plane wave dynamics. Sufficient conditions are provided to avoid the numerical instability of the SSFS method. In contrast to the standard NLS, the plane wave dynamics of the fractional NLS are more chaotic due to the long-range interactions --Abstract, page iii

Numerical algorithms for Schrödinger equation with artificial boundary conditions

We consider a one-dimensional linear Schrödinger problem defined on an infinite domain and approximated by the Crank-Nicolson type finite difference scheme. To solve this problem numerically we restrict the computational domain by introducing the reflective, absorbing or transparent artificial boundary conditions. We investigate the conservativity of the discrete scheme with respect to the mass and energy of the solution. Results of computational experiments are presented and the efficiency of different artificial boundary conditions is discussed

DATA-DRIVEN MODELING AND SIMULATIONS OF SEISMIC WAVES

In recent decades, nonlocal models have been proved to be very effective in the study of complex processes and multiscale phenomena arising in many fields, such as quantum mechanics, geophysics, and cardiac electrophysiology. The fractional Laplacian(−Δ)/2 can be reviewed as nonlocal generalization of the classical Laplacian which has been widely used for the description of memory and hereditary properties of various material and process. However, the nonlocality property of fractional Laplacian introduces challenges in mathematical analysis and computation. Compared to the classical Laplacian, existing numerical methods for the fractional Laplacian still remain limited. The objectives of this research are to develop new numerical methods to solve nonlocal models with fractional Laplacian and apply them to study seismic wave modeling in both homogeneous and heterogeneous media. To this end, we have developed two classes of methods: meshfree pseudospectral method and operator factorization methods. Compared to the current state-of-the-art methods, both of them can achieve higher accuracy with less computational complexity. The operator factorization methods provide a general framework, allowing one to achieve better accuracy with high-degree Lagrange basis functions. The meshfree pseudospectral methods based on global radial basis functions can solve both classical and fractional Laplacians in a single scheme which are the first compatible methods for these two distinct operators. Numerical experiments have demonstrated the effectiveness of our methods on various nonlocal problems. Moreover, we present an extensive study of the variable-order Laplacian operator (−Δ)(x)/2 by using meshfree methods both analytically and numerically. Finally, we apply our numerical methods to solve seismic wave modeling and study the nonlocal effects of fractional wave equation --Abstract, p. i

New Challenges Arising in Engineering Problems with Fractional and Integer Order

Mathematical models have been frequently studied in recent decades, in order to obtain the deeper properties of real-world problems. In particular, if these problems, such as finance, soliton theory and health problems, as well as problems arising in applied science and so on, affect humans from all over the world, studying such problems is inevitable. In this sense, the first step in understanding such problems is the mathematical forms. This comes from modeling events observed in various fields of science, such as physics, chemistry, mechanics, electricity, biology, economy, mathematical applications, and control theory. Moreover, research done involving fractional ordinary or partial differential equations and other relevant topics relating to integer order have attracted the attention of experts from all over the world. Various methods have been presented and developed to solve such models numerically and analytically. Extracted results are generally in the form of numerical solutions, analytical solutions, approximate solutions and periodic properties. With the help of newly developed computational systems, experts have investigated and modeled such problems. Moreover, their graphical simulations have also been presented in the literature. Their graphical simulations, such as 2D, 3D and contour figures, have also been investigated to obtain more and deeper properties of the real world problem

ABC Method and Fractional Momentum Layer for the FDTD Method to Solve the Schrödinger Equation on Unbounded Domains

The finite­difference time­domain (FDTD) method and its generalized variant (G­FDTD) are efficient numerical tools for solving the linear and nonlinear Schrödinger equations because not only are they explicit, allowing parallelization, but they also provide high­order accuracy with relatively inexpensive computational costs. In addition, the G­FDTD method has a relaxed stability condition when compared to the original FDTD method. It is important to note that the existing simulations of the G­FDTD scheme employed analytical solutions to obtain function values at the points along the boundary; however, in simulations for which the analytical solution is unknown, theoretical approximations for values at points along the boundary are desperately needed. Hence, the objective of this dissertation research is to develop absorbing boundary conditions (ABCs) so that the G­FDTD method can be used to solve the nonlinear Schrödinger equation when the analytical solution is unknown. To create the ABCs for the nonlinear Schrödinger equation, we initially determine the associated Engquist­Majda one­way wave equations and then proceed to develop a finite difference scheme for them. These ABCs are made to be adaptive using a windowed Fourier transform to estimate a value of the wavenumber of the carrier wave. These ABCs were tested using the nonlinear Schrödinger equation for 1D and 2D soliton propagation as well as Gaussian packet collision and dipole radiation. Results show that these ABCs perform well, but they have three key limitations. First, there are inherent reflections at the interface of the interior and boundary domains due to the different schemes used the two regions; second, to use the ABCs, one needs to estimate a value for the carrier wavenumber and poor estimates can cause even more reflection at the interface; and finally, the ABCs require different schemes in different regions of the boundary, and this domain decomposition makes the ABCs tedious both to develop and to implement. To address these limitations for the FDTD method, we employ the fractional­order derivative concept to unify the Schrödinger equation with its one­way wave equation over an interval where the fractional order is allowed to vary. Through careful construction of a variable­order fractional momentum operator, outgoing waves may enter the fractionalorder region with little to no reflection and, inside this region, any reflected portions of the wave will decay exponentially with time. The fractional momentum operator is then used to create a fractional­order FDTD scheme. Importantly, this single scheme can be used for the entire computational domain, and the scheme smooths the abrupt transition between the FDTD method and the ABCs. Furthermore, the fractional FDTD scheme relaxes the precision needed for the estimated carrier wavenumber. This fractional FDTD scheme is tested for both the linear and nonlinear Schrödinger equations. Example cases include a 1D Gaussian packet scattering off of a potential, a 1D soliton propagating to the right, as well as 2D soliton propagation, and the collision of Gaussian packets. Results show that the fractional FDTD method outperforms the FDTD method with ABCs

Geometric Integrators for Schrödinger Equations

The celebrated Schrödinger equation is the key to understanding the dynamics of quantum mechanical particles and comes in a variety of forms. Its numerical solution poses numerous challenges, some of which are addressed in this work. Arguably the most important problem in quantum mechanics is the so-called harmonic oscillator due to its good approximation properties for trapping potentials. In Chapter 2, an algebraic correspondence-technique is introduced and applied to construct efficient splitting algorithms, based solely on fast Fourier transforms, which solve quadratic potentials in any number of dimensions exactly - including the important case of rotating particles and non-autonomous trappings after averaging by Magnus expansions. The results are shown to transfer smoothly to the Gross-Pitaevskii equation in Chapter 3. Additionally, the notion of modified nonlinear potentials is introduced and it is shown how to efficiently compute them using Fourier transforms. It is shown how to apply complex coefficient splittings to this nonlinear equation and numerical results corroborate the findings. In the semiclassical limit, the evolution operator becomes highly oscillatory and standard splitting methods suffer from exponentially increasing complexity when raising the order of the method. Algorithms with only quadratic order-dependence of the computational cost are found using the Zassenhaus algorithm. In contrast to classical splittings, special commutators are allowed to appear in the exponents. By construction, they are rapidly decreasing in size with the semiclassical parameter and can be exponentiated using only a few Lanczos iterations. For completeness, an alternative technique based on Hagedorn wavepackets is revisited and interpreted in the light of Magnus expansions and minor improvements are suggested. In the presence of explicit time-dependencies in the semiclassical Hamiltonian, the Zassenhaus algorithm requires a special initiation step. Distinguishing the case of smooth and fast frequencies, it is shown how to adapt the mechanism to obtain an efficiently computable decomposition of an effective Hamiltonian that has been obtained after Magnus expansion, without having to resolve the oscillations by taking a prohibitively small time-step. Chapter 5 considers the Schrödinger eigenvalue problem which can be formulated as an initial value problem after a Wick-rotating the Schrödinger equation to imaginary time. The elliptic nature of the evolution operator restricts standard splittings to low order, ¿ < 3, because of the unavoidable appearance of negative fractional timesteps that correspond to the ill-posed integration backwards in time. The inclusion of modified potentials lifts the order barrier up to ¿ < 5. Both restrictions can be circumvented using complex fractional time-steps with positive real part and sixthorder methods optimized for near-integrable Hamiltonians are presented. Conclusions and pointers to further research are detailed in Chapter 6, with a special focus on optimal quantum control.Bader, PK. (2014). Geometric Integrators for Schrödinger Equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/38716TESISPremios Extraordinarios de tesis doctorale

Numerical and Analytical Methods in Electromagnetics

Like all branches of physics and engineering, electromagnetics relies on mathematical methods for modeling, simulation, and design procedures in all of its aspects (radiation, propagation, scattering, imaging, etc.). Originally, rigorous analytical techniques were the only machinery available to produce any useful results. In the 1960s and 1970s, emphasis was placed on asymptotic techniques, which produced approximations of the fields for very high frequencies when closed-form solutions were not feasible. Later, when computers demonstrated explosive progress, numerical techniques were utilized to develop approximate results of controllable accuracy for arbitrary geometries. In this Special Issue, the most recent advances in the aforementioned approaches are presented to illustrate the state-of-the-art mathematical techniques in electromagnetics