arXiv subjects

Wim Vanroose

Publications and source records attributed to Wim Vanroose.

45 records · Page 3Linked to original sources

arXiv2011

Analyzing the wave number dependency of the convergence rate of a multigrid preconditioned Krylov method for the Helmholtz equation with an absorbing layer

This paper analyzes the Krylov convergence rate of a Helmholtz problem preconditioned with Multigrid. The multigrid method is applied to the Helmholtz problem formulated on a complex contour and uses GMRES as a smoother substitute at each level. A one-dimensional model is analyzed both in a continuous and discrete way. It is shown that the Krylov convergence rate of the continuous problem is independent of the wave number. The discrete problem, however, can deviate significantly from this bound due to a pitchfork in the spectrum. It is further shown in numerical experiments that the convergence rate of the Krylov method approaches the continuous bound as the grid distance $h$ gets small.

math.NA↗

arXiv2011

A preconditioned iterative solver for the scattering solutions of the Schrödinger equation

The Schrödinger equation defines the dynamics of quantum particles which has been an area of unabated interest in physics. We demonstrate how simple transformations of the Schrödinger equation leads to a coupled linear system, whereby each diagonal block is a high frequency Helmholtz problem. Based on this model, we derive indefinite Helmholtz model problems with strongly varying wavenumbers. We employ the iterative approach for their solution. In particular, we develop a preconditioner that has its spectrum restricted to a quadrant (of the complex plane) thereby making it easily invertible by multigrid methods with standard components. This multigrid preconditioner is used in conjuction with suitable Krylov-subspace methods for solving the indefinite Helmholtz model problems. The aim of this study is to report the feasbility of this preconditioner for the model problems. We compare this idea with the other prevalent preconditioning ideas, and discuss its merits. Results of numerical experiments are presented, which complement the proposed ideas, and show that this preconditioner may be used in an automatic setting.

math.NA↗

arXiv2011

Numerical Continuation of Bound and Resonant States of the Two Channel Schrödinger Equation

Resonant solutions of the quantum Schrödinger equation occur at complex energies where the S-matrix becomes singular. Knowledge of such resonances is important in the study of the underlying physical system. Often the Schrödinger equation is dependent on some parameter and one is interested in following the path of the resonances in the complex energy plane as the parameter changes. This is particularly true in coupled channel systems where the resonant behavior is highly dependent on the strength of the channel coupling, the energy separation of the channels and other factors. In previous work it was shown that numerical continuation, a technique familiar in the study of dynamical systems, can be brought to bear on the problem of following the resonance path in one dimensional problems and multi-channel problems without energy separation between the channels. A regularization can be defined that eliminates coalescing poles and zeros that appear in the S-matrix at the origin due to symmetries. Following the zeros of this regularized function then traces the resonance path. In this work we show that this approach can be extended to channels with energy separation, albeit limited to two channels. The issue here is that the energy separation introduces branch cuts in the complex energy domain that need to be eliminated with a so-called uniformization. We demonstrate that the resulting approach is suitable for investigating resonances in two-channel systems and provide an extensive example.

math-ph↗

arXiv2010

Numerical Continuation of resonances and bound states in coupled channel Schrödinger equations

In this contribution, we introduce numerical continuation methods and bifurcation theory, techniques which find their roots in the study of dynamical systems, to the problem of tracing the parameter dependence of bound and resonant states of the quantum mechanical Schrödinger equation. We extend previous work on the subject to systems of coupled equations. Bound and resonant states of the Schrödinger equation can be determined through the poles of the S-matrix, a quantity that can be derived from the asymptotic form of the wave function. We introduce a regularization procedure that essentially transforms the S-matrix into its inverse and improves its smoothness properties, thus making it amenable to numerical continuation. This allows us to automate the process of tracking bound and resonant states when parameters in the Schrödinger equation are varied. We have applied this approach to a number of model problems with satisfying results.

math-ph↗

arXiv2010

Applying numerical continuation to the parameter dependence of solutions of the Schrödinger equation

In molecular reactions at the microscopic level the appearance of resonances has an important influence on the reactivity. It is important to predict when a bound state transitions into a resonance and how these transitions depend on various system parameters such as internuclear distances. The dynamics of such systems are described by the time-independent Schrödinger equation and the resonances are modeled by poles of the S-matrix. Using numerical continuation methods and bifurcation theory, techniques which find their roots in the study of dynamical systems, we are able to develop efficient and robust methods to study the transitions of bound states into resonances. By applying Keller's Pseudo-Arclength continuation, we can minimize the numerical complexity of our algorithm. As continuation methods generally assume smooth and well-behaving functions and the S-matrix is neither, special care has been taken to ensure accurate results. We have successfully applied our approach in a number of model problems involving the radial Schrödinger equation.

math-ph↗

arXiv2010

On the indefinite Helmholtz equation: complex stretched absorbing boundary layers, iterative analysis, and preconditioning

This paper studies and analyzes a preconditioned Krylov solver for Helmholtz problems that are formulated with absorbing boundary layers based on complex coordinate stretching. The preconditioner problem is a Helmholtz problem where not only the coordinates in the absorbing layer have an imaginary part, but also the coordinates in the interior region. This results into a preconditioner problem that is invertible with a multigrid cycle. We give a numerical analysis based on the eigenvalues and evaluate the performance with several numerical experiments. The method is an alternative to the complex shifted Laplacian and it gives a comparable performance for the studied model problems.

math.NA↗

arXiv2007

Coarse-grained analysis of a lattice Boltzmann model for planar streamer fronts

We study the traveling wave solutions of a lattice Boltzmann model for the planar streamer fronts that appear in the transport of electrons through a gas in a strong electrical field. To mimic the physical properties of the impact ionization reaction, we introduce a reaction matrix containing reaction rates that depend on the electron velocities. Via a Chapman--Enskog expansion, one is able to find only a rough approximation for a macroscopic evolution law that describes the traveling wave solution. We propose to compute these solutions with the help of a coarse-grained time-stepper, which is an effective evolution law for the macroscopic fields that only uses appropriately initialized simulations of the lattice Boltzmann model over short time intervals. The traveling wave solution is found as a fixed point of the sequential application of the coarse-grained time-stepper and a shift-back operator. The fixed point is then computed with a Newton-Krylov Solver. We compare the resulting solutions with those of the approximate PDE model, and propose a method to find the minimal physical wave speed.

physics.comp-ph↗

arXiv2006

Coarse-grained computation of traveling waves of lattice Boltzmann models with Newton-Krylov solvers

For many complex dynamical systems, a separation of scales prevails between the (microscopic) level of description of the available model, and the (macroscopic) level at which one would like to observe and analyze the system. For this type of problems, an ``equation-free'' framework has recently been proposed. Using appropriately initialized microscopic simulations, one can build a coarse-grained time-stepper to approximate a time-stepper for the unavailable macroscopic model. Here, we show how one can use this coarse-grained time-stepper to compute coarse-grained traveling wave solutions of a lattice Boltzmann model. In a moving frame, emulated by performing a shift-back operation after the coarse-grained time-step, the traveling wave appears as a steady state, which is computed using an iterative method, such as Newton--GMRES. To accelerate convergence of the GMRES procedure, a macroscopic model-based preconditioner is used, which is derived from the lattice Boltzmann model using a Chapman--Enskog expansion. We illustrate the approach on a lattice Boltzmann model for the Fisher equation and on a model for ionization waves.

physics.comp-ph↗

arXiv2001

Interference of resonances and the double-pole of the S-matrix in a double well system

We discuss the interaction between two resonant states in a quantum double-well structure. The behaviour of the resonant states depends on the coupling between the wells, i.e. the height and width of the barrier that separates them. We distinguish a region with resonant tunneling and a region where the two resonances repel each other. The transition between the two regions is marked by a double pole of the S-matrix.

quant-ph↗