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

Unconditional well-posedness and IMEX improvement of a family of predictor-corrector methods in micromagnetics

Recently, Kim & Wilkening (Convergence of a mass-lumped finite element method for the Landau-Lifshitz equation, Quart. Appl. Math., 76, 383-405, 2018) proposed two novel predictor-corrector methods for the Landau-Lifshitz-Gilbert equation (LLG) in micromagnetics, which models the dynamics of the magnetization in ferromagnetic materials. Both integrators are based on the so-called Landau-Lifshitz form of LLG, use mass-lumped variational formulations discretized by first-order finite elements, and only require the solution of linear systems, despite the nonlinearity of LLG. The first(-order in time) method combines a linear update with an explicit projection of an intermediate approximation onto the unit sphere in order to fulfill the LLG-inherent unit-length constraint at the discrete level. In the second(-order in time) integrator, the projection step is replaced by a linear constraint-preserving variational formulation. In this paper, we extend the analysis of the integrators by proving unconditional well-posedness and by establishing a close connection of the methods with other approaches available in the literature. Moreover, the new analysis also provides a well-posed integrator for the Schrödinger map equation (which is the limit case of LLG for vanishing damping). Finally, we design an implicit-explicit strategy for the treatment of the lower-order field contributions, which significantly reduces the computational cost of the schemes, while preserving their theoretical properties

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

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

Real-Time Complex Langevin - A Differential Programming Perspective

In this thesis, I aim to find solutions to the NP-hard sign-problem that arises when modeling strongly correlated systems in real-time. I will use the complex Langevin (CLE) method, and address its problem of runaway trajectories and incorrect convergence using an implicit solver and a novel kernel optimization scheme, respectively. The implicit solver stabilizes the numerical solution, making the runaway solution problem a thing of the past. It also acts as a regulator, allowing for simulation along the canonical Schwinger-Keldysh contour. Additionally, our investigation shows that a kernel can act as a regulator as well, resulting in an effective change in the action and integral measure while leaving the path integral measure intact. To restore correct convergence in CLE simulations, we present a novel strategy that involves learning a kernel and utilizing functionals that encode relevant prior information, such as symmetries or Euclidean correlator data. Our approach recovers the correct convergence in the non-interacting theory on the Schwinger-Keldysh contour for any real-time extent. It achieves the correct convergence up to three times the real-time extent of the previous benchmark study for the strongly coupled quantum anharmonic oscillator. Furthermore, we investigate the stability of the CLE by calculating the Lyapunov exponents of the CLE and uncovering that the real-time CLE behaves like a chaotic dynamical system. This has consequences for obtaining a reliable gradient of a loss function that contains a real-time CLE simulation. To address this issue, we adapt the shadowing sensitivity method to a stochastic differential equation (SDE), which allows for calculating a reliable gradient of chaotic SDEs

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Uniformly Accurate Methods for Klein-Gordon type Equations

The main contribution of this thesis is the development of a novel class of uniformly accurate methods for Klein-Gordon type equations. Klein-Gordon type equations in the non-relativistic limit regime, i.e., $c\gg 1$, are numerically very challenging to treat, since the solutions are highly oscillatory in time. Standard Gautschi-type methods suffer from severe time step restrictions as they require a CFL-condition $c^2\tau<1$ with time step size $\tau$ to resolve the oscillations. Within this thesis we overcome this difficulty by introducing limit integrators, which allows us to reduce the highly oscillatory problem to the integration of a non-oscillatory limit system. This procedure allows error bounds of order $\mathcal{O}(c^{-2}+\tau^2)$ without any step size restrictions. Thus, these integrators are very efficient in the regime $c\gg 1$. However, limit integrators fail for small values of $c$. In order to derive numerical schemes that work well for small as well as for large $c$, we use the ansatz of "twisted variables", which allows us to develop uniformly accurate methods with respect to $c$. In particular, we introduce efficient and robust uniformly accurate exponential-type integrators which resolve the solution in the relativistic regime as well as in the highly oscillatory non-relativistic regime without any step size restriction. In contrast to previous works, we do not employ any asymptotic nor multiscale expansion of the solution. Compared to classical methods our new schemes allow us to reduce the regularity assumptions as they converge under the same regularity assumptions required for the integration of the corresponding limit system. In addition, the newly derived first- and second-order exponential-type integrators converge to the classical Lie and Strang splitting schemes for the limit system. Moreover, we present uniformly accurate schemes for the Klein-Gordon-Schrödinger and the Klein-Gordon-Zakharov system. For all uniformly accurate integrators we establish rigorous error estimates and underline their uniform convergence property numerically