Local-in-time structure-preserving finite-element schemes for the Euler-Poisson equations

We discuss structure-preserving numerical discretizations for repulsive and attractive Euler-Poisson equations that find applications in fluid-plasma and self-gravitation modeling, respectively. The scheme is fully discrete and structure preserving in the sense that it maintains a discrete energy law, as well as hyperbolic invariant domain properties, such as positivity of the density and a minimum principle of the specific entropy. A detailed discussion of algorithmic details is given, as well as proofs of the claimed properties. We present computational experiments corroborating our analytical findings and demonstrating the computational capabilities of the scheme

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

Modeling Optical Technologies with Continuous and Discrete Nonlinear Schrödinger Equations

This thesis considers continuous and discrete models that describe optical technologies. In the continuous setting, the interaction of modulated pulses in nonlinear dispersive media is addressed. It is shown that there is almost no interaction of two modulating pulse solutions with different carrier waves, regardless of the envelope type. In the discrete setting, the existence, stability, and bifurcation structure of localized solutions are examined in one and two dimensions

Numerical Integrators for Maxwell-Klein-Gordon and Maxwell-Dirac Systems in Highly to Slowly Oscillatory Regimes

Maxwell-Klein-Gordon (MKG) and Maxwell-Dirac (MD) systems physically describe the mutual interaction of moving relativistic particles with their self-generated electromagnetic field. Solving these systems in the nonrelativistic limit regime, i.e. when the speed of light $c$ formally tends to infinity, is numerically very delicate as the solution becomes highly oscillatory in time. In order to resolve the oscillations, standard time integrations schemes require severe restrictions on the time step $\tau\sim c^{-2}$ depending on the small parameter $c^{-2}$ which leads to high computational costs. Within this thesis we propose and analyse two types of numerical integrators to efficiently integrate the MKG and MD systems in highly oscillatory nonrelativistic limit regimes to slowly oscillatory relativistic regimes. The idea for the first type relies on asymptotically expanding the exact solution in the small parameter $c^{-1}$. This results in non-oscillatory Schrödinger-Poisson (SP) limit systems which can be solved efficiently by using classical splitting schemes. We will see that standard Strang splitting schemes, applied to the latter SP systems with step size $\tau$, allow error bounds of order $\mathcal{O}(\tau^2+c^{-N})$ for $N\in \mathbb N$ without any time step restriction. Thus, in the nonrelativistic limit regime $c\rightarrow\infty$ these methods are very efficient and allow an accurate approximation to the exact solution. The second type of numerical integrator is based on "twisted variables" which have been originally introduced for the Klein-Gordon equation in [Baumstark/Faou/Schratz, 2017]. In the case of MKG and MD systems however, due to the strong nonlinear coupling between the components of the solution, the construction and analysis is much more involved. We thereby exploit the main advantage of the "twisted variables" that they have bounded derivatives with respect to $c\rightarrow\infty$. Together with a splitting approach, this allows us to construct an exponential-type splitting method which is first order accurate in time uniformly in $c$. Due to error bounds of order $\mathcal{O}(\tau)$ independent of $c$ without any restriction on the time step $\tau$, these schemes are efficient in highly to slowly oscillatory regimes

Potential-based Formulations of the Navier-Stokes Equations and their Application

Based on a Clebsch-like velocity representation and a combination of classical variational principles for the special cases of ideal and Stokes flow a novel discontinuous Lagrangian is constructed; it bypasses the known problems associated with non-physical solutions and recovers the classical Navier-Stokes equations together with the balance of inner energy in the limit when an emerging characteristic frequency parameter tends to infinity. Additionally, a generalized Clebsch transformation for viscous flow is established for the first time. Next, an exact first integral of the unsteady, three-dimensional, incompressible Navier-Stokes equations is derived; following which gauge freedoms are explored leading to favourable reductions in the complexity of the equation set and number of unknowns, enabling a self-adjoint variational principle for steady viscous flow to be constructed. Concurrently, appropriate commonly occurring physical and auxiliary boundary conditions are prescribed, including establishment of a first integral for the dynamic boundary condition at a free surface. Starting from this new formulation, three classical flow problems are considered, the results obtained being in total agreement with solutions in the open literature. A new least-squares finite element method based on the first integral of the steady two-dimensional, incompressible, Navier-Stokes equations is developed, with optimal convergence rates established theoretically. The method is analysed comprehensively, thoroughly validated and shown to be competitive when compared to a corresponding, standard, primitive-variable, finite element formulation. Implementation details are provided, and the well-known problem of mass conservation addressed and resolved via selective weighting. The attractive positive definiteness of the resulting linear systems enables employment of a customized scalable algebraic multigrid method for efficient error reduction. The solution of several engineering related problems from the fields of lubrication and film flow demonstrate the flexibility and efficiency of the proposed method, including the case of unsteady flow, while revealing new physical insights of interest in their own right

Simulations of Critical Currents in Polycrystalline Superconductors Using Time-Dependent Ginzburg–Landau Theory

In this thesis, we investigate the in-field critical current density $J_\text{c} (B)$ of polycrystalline superconducting systems with grain boundaries modelled as Josephson-type planar defects, both analytically and through computational time-dependent Ginzburg--Landau (TDGL) simulations in 2D and 3D. For very narrow SNS Josephson junctions (JJs), with widths smaller than the superconducting coherence length, we derive what to our knowledge are the first analytic expressions for $J_\text{c} (B)$ across a JJ over the entire applied magnetic field range. We extend the validity of our analytic expressions to describe wider junctions and confirm them using TDGL simulations. We model superconducting systems containing grain boundaries as a network of JJs by using large-scale 3D TDGL simulations applying state-of-the-art solvers implemented on GPU architectures. These simulations of $J_\text{c} (B)$ have similar magnitudes and dependencies on applied magnetic field to those observed experimentally in optimised commercial superconductors. They provide an explanation for the $B^{-0.6}$ dependence found for $J_\text{c} (B)$ in high temperature superconductors and are the first to correctly provide the inverse power-law grain size behaviour as well as the Kramer field dependence, widely found in many low temperature superconductors

The Maxwell-Landau-Lifshitz-Gilbert System: Mathematical Theory and Numerical Approximation

This thesis deals with the mathematical theory and numerical approximation of the Landau--Lifshitz--Gilbert equation coupled to the Maxwell equations without artificial boundary conditions. As a starting point, the physical equations are stated on the unbounded three dimensional space and reformulated in a mathematically precise way to a coupled partial differential -- boundary integral system. We derive a weak form of the whole coupled system, state the relation to the strong form and show uniqueness of the Maxwell part of the solution. A numerical algorithm is proposed based on the tangent plane scheme for the LLG part and using a finite element and boundary element coupling as spatial discretization and the backward Euler method and Convolution Quadrature as time discretization for the interior Maxwell part and the boundary, respectively. Under minimal assumptions on the regularity of solutions, we present well-posedness and convergence of the numerical algorithm. For the pure Maxwell equations without the coupling to the LLG equation, we are able to show stronger results than in the coupled case. We derive a weak form for the Maxwell transmission problem and demonstrate existence and uniqueness of the weak solutions as well as equivalence with a strong solution. The proposed algorithm of finite-element/ boundary-element coupling via Convolution Quadrature converges with only minimal assumptions on the regularity of the input data. Again for the full Maxwell--LLG system, we show a-priori error bounds in the situation of a sufficiently regular solution. This is done by a combination of the known linearly implicit backward difference formula time discretizations with higher order non-conforming finite element space discretizations for the LLG equation and the leapfrog and Convolution Quadrature time discretization with higher order discontinuous Galerkin elements and continuous boundary elements for the boundary integral formulation of Maxwell\u27s equations. The precise method of coupling allows us to solve the system at the cost of the individual parts, with the same convergence rates under the same regularity assumptions and the same CFL conditions as for an uncoupled examination. Numerical experiments illustrate and expand on the theoretical results and demonstrate the applicability of the methods. For the formulation of the boundary integral equations, the study of the Laplace transform is inevitable. We collect and extend the properties of the Laplace transform from literature. In the suitable functional analytic setting, we give extensive proofs in a self contained way of all the required properties