Numerical solution of the complex Ginzburg-Landau equation alternatives for its time integration

Treball de Fi de Màster Universitari en Matemàtica Computacional (Pla de 2013). Codi: SIQ527. Curs 2021/2022 (A distància)La ecuación Ginzburg-Landau compleja en su forma cúbica describe una amplia gama de fenómenos para muchos sistemas de la física y presenta muchas estructuras coherentes estables entre sus soluciones. De ahí que se haya estudiado intensamente a lo largo de los años y que sea un laboratorio ideal para aprender sobre los esquemas de integración temporal. Desde el punto de vista numérico se pueden utilizar métodos pseudoespectrales para resolver la discretización espacial, hay que resolverlo con variable compleja y se pueden utilizar métodos específicos para la integración temporal como la Exponential Time Differencing o Integrating Factor Methods gracias a la presencia de términos lineales y no lineales integrables. El objetivo de esta tesis es comparar algunos de estos métodos analizando su orden y eficiencia. Para ello, se han programado los diferentes esquemas en Fortran y se han validado sus resultados mediante una pila de casos de prueba. A continuación, utilizando uno de ellos como referencia y calculando la solución de una prueba con un paso de tiempo muy pequeño, se han comparado los demás. Entre otros resultados, se ha constatado la buena convergencia de los métodos de orden superior o la mejor velocidad de los métodos que integran analíticamente parte de la ecuación o que necesitan menos transformadas de Fourier

Coupled oscillators with power-law interaction and their fractional dynamics analogues

The one-dimensional chain of coupled oscillators with long-range power-law interaction is considered. The equation of motion in the infrared limit are mapped onto the continuum equation with the Riesz fractional derivative of order $\alpha$, when $0<\alpha<2$. The evolution of soliton-like and breather-like structures are obtained numerically and compared for both types of simulations: using the chain of oscillators and using the continuous medium equation with the fractional derivative.Comment: 16 pages, 5 figure

A fast two-level Strang splitting method for multi-dimensional spatial fractional Allen-Cahn equations with discrete maximum principle

In this paper, we study the numerical solutions of the multi-dimensional spatial fractional Allen-Cahn equations. After semi-discretization for the spatial fractional Riesz derivative, a system of nonlinear ordinary differential equations with Toeplitz structure is obtained. For the sake of reducing the computational complexity, a two-level Strang splitting method is proposed, where the Toeplitz matrix in the system is split into the sum of a circulant matrix and a skew-circulant matrix. Therefore, the proposed method can be quickly implemented by the fast Fourier transform, substituting to calculate the expensive Toeplitz matrix exponential. Theoretically, the discrete maximum principle of our method is unconditionally preserved. Moreover, the analysis of error in the infinite norm with second-order accuracy is conducted in both time and space. Finally, numerical tests are given to corroborate our theoretical conclusions and the efficiency of the proposed method

Fourier-Splitting methods for the dynamics of rotating Bose-Einstein condensates

We present a new method to propagate rotating Bose-Einstein condensates subject to explicitly time-dependent trapping potentials. Using algebraic techniques, we combine Magnus expansions and splitting methods to yield any order methods for the multivariate and nonautonomous quadratic part of the Hamiltonian that can be computed using only Fourier transforms at the cost of solving a small system of polynomial equations. The resulting scheme solves the challenging component of the (nonlinear) Hamiltonian and can be combined with optimized splitting methods to yield efficient algorithms for rotating Bose-Einstein condensates. The method is particularly efficient for potentials that can be regarded as perturbed rotating and trapped condensates, e.g., for small nonlinearities, since it retains the near-integrable structure of the problem. For large nonlinearities, the method remains highly efficient if higher order p > 2 is sought. Furthermore, we show how it can adapted to the presence of dissipation terms. Numerical examples illustrate the performance of the scheme.Comment: 15 pages, 4 figures, as submitted to journa

On dynamical low-rank integrators for matrix differential equations

This thesis is concerned with dynamical low-rank integrators for matrix differential equations, typically stemming from space discretizations of partial differential equations. We first construct and analyze a dynamical low-rank integrator for second-order matrix differential equations, which is based on a Strang splitting and the projector-splitting integrator, a dynamical low-rank integrator for first-order matrix differential equations proposed by Lubich and Osedelets in 2014. For the analysis, we derive coupled recursive inequalities, where we express the global error of the scheme in terms of a time-discretization error and a low-rank error contribution. The first can be treated with Taylor series expansion of the exact solution. For the latter, we make use of an induction argument and the convergence result derived by Kieri, Lubich, and Walach in 2016 for the projector-splitting integrator. From the original method, several variants are derived which are tailored to, e.g., stiff or highly oscillatory second-order problems. After discussing details on the implementation of dynamical low-rank schemes, we turn towards rank-adaptivity. For the projector-splitting integrator we derive both a technique to realize changes in the approximation ranks efficiently and a heuristic to choose the rank appropriately over time. The core idea is to determine the rank such that the error of the low-rank approximation does not spoil the time-discretization error. Based on the rank-adaptive pendant of the projector-splitting integrator, rank-adaptive dynamical low-rank integrators for (stiff and non-stiff) first-order and second-order matrix differential equations are derived. The thesis is concluded with numerical experiments to confirm our theoretical findings

Schnelle Löser für Partielle Differentialgleichungen

This workshop was well attended by 52 participants with broad geographic representation from 11 countries and 3 continents. It was a nice blend of researchers with various backgrounds

A Space-time Smooth Artificial Viscosity Method For Nonlinear Conservation Laws

We introduce a new methodology for adding localized, space-time smooth, artificial viscosity to nonlinear systems of conservation laws which propagate shock waves, rarefactions, and contact discontinuities, which we call the $C$-method. We shall focus our attention on the compressible Euler equations in one space dimension. The novel feature of our approach involves the coupling of a linear scalar reaction-diffusion equation to our system of conservation laws, whose solution $C(x,t)$ is the coefficient to an additional (and artificial) term added to the flux, which determines the location, localization, and strength of the artificial viscosity. Near shock discontinuities, $C(x,t)$ is large and localized, and transitions smoothly in space-time to zero away from discontinuities. Our approach is a provably convergent, spacetime-regularized variant of the original idea of Richtmeyer and Von Neumann, and is provided at the level of the PDE, thus allowing a host of numerical discretization schemes to be employed. We demonstrate the effectiveness of the $C$-method with three different numerical implementations and apply these to a collection of classical problems: the Sod shock-tube, the Osher-Shu shock-tube, the Woodward-Colella blast wave and the Leblanc shock-tube. First, we use a classical continuous finite-element implementation using second-order discretization in both space and time, FEM-C. Second, we use a simplified WENO scheme within our $C$-method framework, WENO-C. Third, we use WENO with the Lax-Friedrichs flux together with the $C$-equation, and call this WENO-LF-C. All three schemes yield higher-order discretization strategies, which provide sharp shock resolution with minimal overshoot and noise, and compare well with higher-order WENO schemes that employ approximate Riemann solvers, outperforming them for the difficult Leblanc shock tube experiment.Comment: 34 pages, 27 figure