Kurt Lust

An Adaptive Newton--Picard Algorithm with Subspace Iteration for Computing Periodic Solutions

This paper is concerned with the efficient computation of periodic orbits in large-scale dynamical systems that arise after spatial discretization of partial differential equations (PDEs). A hybrid Newton--Picard scheme based on the shooting method is derived, which in its simplest form is the recursive projection method (RPM) of Shroff and Keller [SIAM J. Numer. Anal., 30 (1993), pp. 1099--1120] and is used to compute and determine the stability of both stable and unstable periodic orbits. The number of time integrations needed to obtain a solution is shown to be determined only by the systems dynamics. This contrasts with traditional approaches based on Newtons method, for which the number of time integrations grows with the order of the spatial discretization. Two test examples are given to show the performance of the methods and to illustrate various theoretical points.

Reduction and reconstruction for self-similar dynamical systems

We present a general method for analysing and numerically solving partial differential equations with self-similar solutions. The method employs ideas from symmetry reduction in geometric mechanics, and involves separating the dynamics on the shape space (which determines the overall shape of the solution) from those on the group space (which determines the size and scale of the solution). The method is computationally tractable as well, allowing one to compute self-similar solutions by evolving a dynamical system to a steady state, in a scaled reference frame where the self-similarity has been factored out. More generally, bifurcation techniques can be used to find self-similar solutions, and determine their behaviour as parameters in the equations are varied. The method is given for an arbitrary Lie group, providing equations for the dynamics on the reduced space, for reconstructing the full dynamics and for determining the resulting scaling laws for self-similar solutions. We illustrate the technique with a numerical example, computing self-similar solutions of the Burgers equation.

Improved numerical Floquet multipliers

This paper studies numerical methods for linear stability analysis of periodic solutions in codes for bifurcation analysis of small systems of ordinary differential equations (ODEs). Popular techniques in use today (including the AUTO97 method) produce very inaccurate Floquet multipliers if the system has very large or small multipliers. These codes compute the monodromy matrix explicitly or as a matrix pencil of two matrices. The monodromy matrix arises naturally as a product of many matrices in many numerical methods, but this is not exploited. In this case, all Floquet multipliers can be computed with very high precision by using the periodic Schur decomposition and corresponding algorithm [Bojanczyk et al., 1992]. The time discretisation of the periodic orbit becomes the limiting factor for the accuracy. We present just enough of the numerical methods to show how the Floquet multipliers are currently computed and how the periodic Schur decomposition can be fitted into existing codes but omit all details. However, we show extensive test results for a few artificial matrices and for two four-dimensional systems with some very large and very small Floquet multipliers to illustrate the problems experienced by current techniques and the better results obtained using the periodic Schur decomposition. We use a modified version of AUTO97 [Doedel et al., 1997] in our experiments.

Computation, Continuation and Bifurcation Analysis of Periodic Solutions of Delay Differential Equations

We present a new numerical method for the ecient computation of periodic solutions of nonlinear systems of Delay Dierential Equations (DDEs) with several discrete delays. This method exploits the typical spectral properties of the monodromy matrix of a DDE and allows eective computation of the dominant Floquet multipliers to determine the stability of a periodic solution. We show that the method is particularly suited to trace a branch of periodic solutions using continuation and can be used to locate bifurcation points with good accuracy.

A Newton-Picard shooting method for computing periodic solutions of large-scale dynamical systems

Abstract A numerical method is presented for the efficient computation and continuation of periodic solutions of large systems of ordinary differential equations. The method is particularly useful when a shooting approach based on full Newton or quasi-Newton iteration is prohibitively expensive. Both stable and unstable solutions can be computed. The basic idea is to split the eigenspace of the monodromy matrix around the periodic orbit into two orthogonal components which contain, respectively, all eigenvalues with norm less than or greater than some threshold value less than unity. In the former ‘stable’ subspace one performs time integration, which is equivalent to a Picard iteration; in the much smaller, ‘unstable’ subspace a Newton step is carried out. A strategy for computing the stable and unstable subspaces without forming the full monodromy matrix is briefly discussed. Numerical results are presented for the one-dimensional Brusselator model. The Newton-Picard method with an appropriately chosen threshold value proves to be accurate and much more efficient than shooting based on full Newton iteration.

Run-time load balancing support for a parallel multiblock Euler/Navier-Stokes code with adaptive refinement on distributed memory computers

Abstract This paper describes the parallel implementation of algorithms requiring run-time load redistribution with the aid of the parallel programming library loco . As a typical application, a 2D finite volume multiblock Euler/Navier-Stokes code with block-wise adaptive mesh refinement is discussed. The loco software handles the communication between blocks and the distribution of blocks among the processors, thereby performing automatic load balancing at run-time. The loco library is interfaced with both the native NX communication primitives on Intel iPSC hypercubes and the PVM software on workstation clusters. The parallel performance of the code on the Intel iPSC/860 and on a DEC Alpha workstation cluster is discussed. In particular the effects of mesh refinement on the load balance are investigated.

A computer-assisted study of pulse dynamics in anisotropic media

Abstract This study focuses on the computer-assisted stability analysis of travelling pulse-like structures in spatially periodic heterogeneous reaction–diffusion media. The physical motivation comes from pulse propagation in thin annular domains on a diffusionally anisotropic catalytic surface. The study was performed by computing the travelling pulse-like structures as limit cycles of the spatially discretized PDE, which in turn is performed in two ways: a Newton method based on a pseudospectral discretization of the PDE, and a Newton–Picard method based on a finite difference discretization. Details about the spectra of these modulated pulse-like structures are discussed, including how they may be compared with the spectra of pulses in homogeneous media. The effects of anisotropy on the dynamics of pulses and pulse pairs are studied. Beyond shifting the location of bifurcations present in homogeneous media, anisotropy can also introduce certain new instabilities.

Coarse-Grained Simulation and Bifurcation Analysis Using Microscopic Time-Steppers

In many science and engineering problems, one observes smooth behaviour on macroscopic space and time scales. However, sometimes only a microscopic evolution law is known. In such cases, one can approximate the macroscopic time evolution by performing appropriately initialized simulations of the available microscopic model in small portions of the space-time domain. This coarse-grained time-stepper can be used to perform time-stepper based numerical bifurcation analysis. We discuss our recent results concerning the accuracy of the proposed methods.

Numerical bifurcation analysis of lattice Boltzmann models: A reaction-diffusion example

We study two strategies to perform a time stepper based numerical bifurcation analysis of systems modeled by lattice Boltzmann methods, one using the lattice Boltzmann model as the time stepper and the other the coarse-grained time stepper proposed in Kevrekidis et al., CMS 1(4). We show that techniques developed for time stepper based numerical bifurcation analysis of partial differential equations (PDEs) can be used for lattice Boltzmann models as well. The results for both approaches are also compared with an equivalent PDE description.

Computation and Stability Analysis of Solutions of Periodic Delay Differential Algebraic Equations

This paper is concerned with the numerical computation, continuation and stability analysis of periodic solutions of periodic delay differential algebraic equations. We consider systems with a time-periodic right hand side function and time-periodic delays. We introduce numerical algorithms based on collocation to compute periodic solutions and their stability. The presented methods combine knowledge from numerical methods for delay equations and differential algebraic equations. Our algorithms are illustrated with numerical results for two models.Copyright