Showing papers on "Numerical analysis published in 2003"

Equation-Free, Coarse-Grained Multiscale Computation: Enabling Mocroscopic Simulators to Perform System-Level Analysis

Abstract: We present and discuss a framework for computer-aided multiscale analysis, which enables models at a fine (microscopic/stochastic) level of description to perform modeling tasks at a coarse (macroscopic, systems) level. These macroscopic modeling tasks, yielding information over long time and large space scales, are accomplished through appropriately initialized calls to the microscopic simulator for only short times and small spatial domains. Traditional modeling approaches first involve the derivation of macroscopic evolution equations (balances closed through constitutive relations). An arsenal of analytical and numerical techniques for the efficient solution of such evolution equations (usually Partial Differential Equations, PDEs) is then brought to bear on the problem. Our equation-free (EF) approach, introduced in (1), when successful, can bypass the derivation of the macroscopic evolution equations when these equations conceptually exist but are not available in closed form. We discuss how the mathematics-assisted development of a computational superstructure may enable alternative descriptions of the problem physics (e.g. Lattice Boltzmann (LB), kinetic Monte Carlo (KMC) or Molecular Dynamics (MD) microscopic simulators, executed over relatively short time and space scales) to perform systems level tasks (integration over relatively large time and space scales,"coarse" bifurcation analysis, optimization, and control) directly. In effect, the procedure constitutes a system identification based, "closure-on-demand" computational toolkit, bridging microscopic/stochastic simulation with traditional continuum scientific computation and numerical analysis. We will briefly survey the application of these "numerical enabling technology" ideas through examples including the computation of coarsely self-similar solutions, and discuss various features, limitations and potential extensions of the approach.

Modeling Textures with Total Variation Minimization and Oscillating Patterns in Image Processing

Abstract: This paper is devoted to the modeling of real textured images by functional minimization and partial differential equations. Following the ideas of Yves Meyer in a total variation minimization framework of L. Rudin, S. Osher, and E. Fatemi, we decompose a given (possible textured) image f into a sum of two functions u+v, where u∈BV is a function of bounded variation (a cartoon or sketchy approximation of f), while v is a function representing the texture or noise. To model v we use the space of oscillating functions introduced by Yves Meyer, which is in some sense the dual of the BV space. The new algorithm is very simple, making use of differential equations and is easily solved in practice. Finally, we implement the method by finite differences, and we present various numerical results on real textured images, showing the obtained decomposition u+v, but we also show how the method can be used for texture discrimination and texture segmentation.

Solving ODEs with MATLAB

Abstract: From the Publisher: This book is for people who need to solve ordinary differential equations (ODEs), both initial value problems (IVPs) and boundary value problems (BVPs) as well as delay differential equations (DDEs). These topics are usually taught in separate courses of length one semester each, but solving ODEs with Matlab provides a sound treatment of all three in about 250 pages. The chapters on each of these topics begin with a discussion of "the facts of life" for the problem, mainly by means of examples. Numerical methods for the problem are then developed - but only the methods most widely used. Although the treatment of each method is brief and technical issues are minimized, the issues important in practice and for understanding the codes are discussed. Often solving a real problem is much more than just learning how to call a code. The last part of each chapter is a tutorial that shows how to solve problems by means of small but realistic examples.

Geometric numerical integration illustrated by the Störmer-Verlet method

Abstract: The subject of geometric numerical integration deals with numerical integrators that preserve geometric properties of the flow of a differential equation, and it explains how structure preservation leads to an improved long-time behaviour. This article illustrates concepts and results of geometric numerical integration on the important example of the Stormer/Verlet method. It thus presents a cross-section of the recent monograph by the authors, enriched by some additional material. After an introduction to the Newton-Stormer-Verlet-leapfrog method and its various interpretations, there follows a discussion of geometric properties: reversibility, symplecticity, volume preservation, and conservation of first integrals. The extension to Hamiltonian systems on manifolds is also described. The theoretical foundation relies on a backward error analysis, which translates the geometric properties of the method into the structure of a modified differential equation, whose flow is nearly identical to the numerical method. Combined with results from perturbation theory, this explains the excellent long-time behaviour of the method: long-time energy conservation, linear error growth and preservation of invariant tori in near-integrable systems, a discrete virial theorem, preservation of adiabatic invariants.

Optimization of the split-step Fourier method in modeling optical-fiber communications systems

Abstract: We studied the efficiency of different implementations of the split-step Fourier method for solving the nonlinear Schro/spl uml/dinger equation that employ different step-size selection criteria. We compared the performance of the different implementations for a variety of pulse formats and systems, including higher order solitons, collisions of soliton pulses, a single-channel periodically stationary dispersion-managed soliton system, and chirped return to zero systems with single and multiple channels. We introduce a globally third-order accurate split-step scheme, in which a bound on the local error is used to select the step size. In many cases, this method is the most efficient when compared with commonly used step-size selection criteria, and it is robust for a wide range of systems providing a system-independent rule for choosing the step sizes. We find that a step-size selection method based on limiting the nonlinear phase rotation of each step is not efficient for many optical-fiber transmission systems, although it works well for solitons. We also tested a method that uses a logarithmic step-size distribution to bound the amount of spurious four-wave mixing. This method is as efficient as other second-order schemes in the single-channel dispersion-managed soliton system, while it is not efficient in other cases including multichannel simulations. We find that in most cases, the simple approach in which the step size is held constant is the least efficient of all the methods. Finally, we implemented a method in which the step size is inversely proportional to the largest group velocity difference between channels. This scheme performs best in multichannel optical communications systems for the values of accuracy typically required in most transmission simulations.

Adaptive low-rank approximation of collocation matrices

Abstract: This article deals with the solution of integral equations using collocation methods with almost linear complexity. Methods such as fast multipole, panel clustering and H-matrix methods gain their efficiency from approximating the kernel function. The proposed algorithm which uses the H-matrix format, in contrast, is purely algebraic and relies on a small part of the collocation matrix for its blockwise approximation by low-rank matrices. Furthermore, a new algorithm for matrix partitioning that significantly reduces the number of blocks generated is presented.

Fast Sweeping Algorithms for a Class of Hamilton--Jacobi Equations

Abstract: We derive a Godunov-type numerical flux for the class of strictly convex, homogeneous Hamiltonians that includes $H(p,q)=\sqrt{ap^{2}+bq^{2}-2cpq},$ $c^{2}

Poincaré-Friedrichs Inequalities for Piecewise H 1 Functions

Abstract: Poincare--Friedrichs inequalities for piecewise H1 functions are established. They can be applied to classical nonconforming finite element methods, mortar methods, and discontinuous Galerkin methods.

Higher-Order Numerical Methods for Transient Wave Equations

Abstract: Solving efficiently the wave equations involved in modeling acoustic, elastic or electromagnetic wave propagation remains a challenge both for research and industry. To attack the problems coming from the propagative character of the solution, the author constructs higher-order numerical methods to reduce the size of the meshes, and consequently the time and space stepping, dramatically improving storage and computing times. This book surveys higher-order finite difference methods and develops various mass-lumped finite (also called spectral) element methods for the transient wave equations, and presents the most efficient methods, respecting both accuracy and stability for each sort of problem. A central role is played by the notion of the dispersion relation for analyzing the methods. The last chapter is devoted to unbounded domains which are modeled using perfectly matched layer (PML) techniques. Numerical examples are given.

One-dimensional models for blood flow in arteries

Abstract: In this paper a family of one-dimensional nonlinear systems which model the blood pulse propagation in compliant arteries is presented and investigated. They are obtained by averaging the Navier-Stokes equation on each section of an arterial vessel and using simplified models for the vessel compliance. Different differential operators arise depending on the simplifications made on the structural model. Starting from the most basic assumption of pure elastic instantaneous equilibrium, which provides a well-known algebraic relation between intramural pressure and vessel section area, we analyse in turn the effects of terms accounting for inertia, longitudinal pre-stress and viscoelasticity. The problem of how to account for branching and possible discontinuous wall properties is addressed, the latter aspect being relevant in the presence of prosthesis and stents. To this purpose a domain decomposition approach is adopted and the conditions which ensure the stability of the coupling are provided. The numerical method here used in order to carry out several test cases for the assessment of the proposed models is based on a finite element Taylor-Galerkin scheme combined with operator splitting techniques.

On an explicit finite difference method for fractional diffusion equations

Abstract: A numerical method to solve the fractional diffusion equation, which could also be easily extended to many other fractional dynamics equations, is considered. These fractional equations have been proposed in order to describe anomalous transport characterized by non-Markovian kinetics and the breakdown of Fick's law. In this paper we combine the forward time centered space (FTCS) method, well known for the numerical integration of ordinary diffusion equations, with the Grunwald-Letnikov definition of the fractional derivative operator to obtain an explicit fractional FTCS scheme for solving the fractional diffusion equation. The resulting method is amenable to a stability analysis a la von Neumann. We show that the analytical stability bounds are in excellent agreement with numerical tests. Comparison between exact analytical solutions and numerical predictions are made.

New Cartesian grid methods for interface problems using the finite element formulation

Abstract: New finite element methods based on Cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients. The triangulations in these methods do not need to fit the interfaces. The basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately. Both non-conforming and conforming finite element spaces are considered. Corresponding interpolation functions are proved to be second order accurate in the maximum norm. The conforming finite element method has been shown to be convergent. With Cartesian triangulations, these new methods can be used as finite difference methods. Numerical examples are provided to support the methods and the theoretical analysis.

Ordered Upwind Methods for Static Hamilton--Jacobi Equations: Theory and Algorithms

Abstract: We develop a family of fast methods for approximating the solutions to a wide class of static Hamilton--Jacobi PDEs; these fast methods include both semi-Lagrangian and fully Eulerian versions. Numerical solutions to these problems are typically obtained by solving large systems of coupled nonlinear discretized equations. Our techniques, which we refer to as "Ordered Upwind Methods" (OUMs), use partial information about the characteristic directions to decouple these nonlinear systems, greatly reducing the computational labor. Our techniques are considered in the context of control-theoretic and front-propagation problems. We begin by discussing existing OUMs, focusing on those designed for isotropic problems. We then introduce a new class of OUMs which decouple systems for general (anisotropic) problems. We prove convergence of one such scheme to the viscosity solution of the corresponding Hamilton--Jacobi PDE. Next, we introduce a set of finite-differences methods based on an analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems. The performance of the methods is analyzed, and computational experiments are performed using test problems from computational geometry and seismology.

Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements

Abstract: In computation of fluid-structure interactions, we use mesh update methods consisting of mesh-moving and remeshing-as-needed. When the geometries lire complex and the structural displacements are large, it becomes even more important that the mesh moving techniques lire designed with the objective to reduce the frequency of remeshing. To that end, we present here mesh moving techniques where the motion of the nodes is governed by the equations of elasticity, with selective treatment of mesh deformation based on element sizes as well as deformation modes in terms of shape and volume changes. We also present results from application of these techniques to a set of two-dimensional test cases.

Comparison of Several Difference Schemes on 1D and 2D Test Problems for the Euler Equations

Abstract: The results of computations with eight explicit finite difference schemes on a suite of one-dimensional and two-dimensional test problems for the Euler equations are presented in various formats. Both dimensionally split and two-dimensional schemes are represented, as are central and upwind-biased methods, and all are at least second-order accurate.

An Immersed Interface Method for Incompressible Navier-Stokes Equations

Abstract: The method developed in this paper is motivated by Peskin's immersed boundary (IB) method, and allows one to model the motion of flexible membranes or other structures immersed in viscous incompressible fluid using a fluid solver on a fixed Cartesian grid. The IB method uses a set of discrete delta functions to spread the entire singular force exerted by the immersed boundary to the nearby fluid grid points. Our method instead incorporates part of this force into jump conditions for the pressure, avoiding discrete dipole terms that adversely affect the accuracy near the immersed boundary. This has been implemented for the two-dimensional incompressible Navier--Stokes equations using a high-resolution finite-volume method for the advective terms and a projection method to enforce incompressibility. In the projection step, the correct jump in pressure is imposed in the course of solving the Poisson problem. This gives sharp resolution of the pressure across the interface and also gives better volume conservation than the traditional IB method. Comparisons between this method and the IB method are presented for several test problems. Numerical studies of the convergence and order of accuracy are included.

Algorithm Developments for Discrete Adjoint Methods

Abstract: This paper presents a number of algorithm developments for adjoint methods using the 'discrete' approach in which the discretisation of the non-linear equations is linearised and the resulting matrix is then transposed. With a new iterative procedure for solving the adjoint equations, exact numerical equivalence is maintained between the linear and adjoint discretisations. The incorporation of strong boundary conditions within the discrete approach is discussed, as well as a new application of adjoint methods to linear unsteady flow in turbomachinery.

Estimating differential quantities using polynomial fitting of osculating jets

Abstract: This paper addresses the pointwise estimation of differential properties of a smooth manifold S---a curve in the plane or a surface in 3D--- assuming a point cloud sampled over S is provided. The method consists of fitting the local representation of the manifold using a jet, by either interpolating or approximating. A jet is a truncated Taylor expansion, and the incentive for using jets is that they encode all local geometric quantities---such as normal or curvatures.On the way to using jets, the question of estimating differential properties is recasted into the more general framework of multivariate interpolation/approximation, a well-studied problem in numerical analysis. On a theoretical perspective, we prove several convergence results when the samples get denser. For curves and surfaces, these results involve asymptotic estimates with convergence rates depending upon the degree of the jet used. For the particular case of curves, an error bound is also derived. To the best of our knowledge, these results are among the first ones providing accurate estimates for differential quantities of order three and more. On the algorithmic side, we solve the interpolation/approximation problem using Vandermonde systems. Experimental results for surface of R3 are reported. These experiments illustrate the asymptotic convergence results, but also the robustness of the methods on general Computer Graphics models.

Adaptive Mesh Methods for One- and Two-Dimensional Hyperbolic Conservation Laws

Abstract: We develop efficient moving mesh algorithms for one- and two-dimensional hyperbolic systems of conservation laws. The algorithms are formed by two independent parts: PDE evolution and mesh-redistribution. The first part can be any appropriate high-resolution scheme, and the second part is based on an iterative procedure. In each iteration, meshes are first redistributed by an equidistribution principle, and then on the resulting new grids the underlying numerical solutions are updated by a conservative-interpolation formula proposed in this work. The iteration for the mesh-redistribution at a given time step is complete when the meshes governed by a nonlinear equation reach the equilibrium state. The main idea of the proposed method is to keep the mass-conservation of the underlying numerical solution at each redistribution step. In one dimension, we can show that the underlying numerical approximation obtained in the mesh-redistribution part satisfies the desired TVD property, which guarantees that the numerical solution at any time level is TVD, provided that the PDE solver in the first part satisfies such a property. Several test problems in one and two dimensions are computed using the proposed moving mesh algorithm. The computations demonstrate that our methods are efficient for solving problems with shock discontinuities, obtaining the same resolution with a much smaller number of grid points than the uniform mesh approach.

The Scaled Boundary Finite Element Method

Abstract: Foreword. Preface. Fundamentals of Numerical Analysis. Novel Computational Procedure. PART I: MODEL PROBLEM: LINE ELEMENT FOR SCALAR WAVE EQUATION. Concepts of Scaled Boundary Transformation of Geometry and Similarity. Wedge and Truncated Semi-Infinite Wedge of Shear Plate. Scaled Boundary Transformation-Based Derivation. Mechanically-Based Derivation. Modelisation with Single Line Finite Element. Statics. Mass of Wedge. High-Frequency Asymptotic Expansion for Dynamic Stiffness of Truncated Semi-Infinite Wedge. Numerical Solution of Dynamic Stiffness, Unit-Impulse Response and Displacement of Truncated Semi-Infinite Wedge. Analytical Solution in Frequency Domain. Implementation. Conclusions. Appendix A: Solid Modelling. Appendix B: Harmonic Motion and Fourier Transformation. Appendix C: Dynamic Unbounded Medium-Structure Interaction. Appendix D: Historical Note. PART II: TWO- AND THREE-DIMENSIONAL ELASTODYNAMICS, STATICS AND DIFFUSION. Fundamental Equations. Statics. Mass Matrix of Bounded Medium. High-Frequency Asymptotic Expansion for Dynamic Stiffness of Unbounded Medium. Numerical Solution of Dynamic Stiffness, Unit-Impulse Response and Displacement of Unbounded Medium. Analytical Solution in Frequency Domain. Extensions. Substructuring. Examples for Bounded Media. Examples for Unbounded Media. Error Estimation and Adaptivity. Concluding Remarks. References. Index.

Korn's inequalities for piecewise $H^1$ vector fields

Abstract: Korn's inequalities for piecewise H 1 vector fields are established. They can be applied to classical nonconforming finite element methods, mortar methods and discontinuous Galerkin methods.

A finite element method for domain decomposition with non-matching grids

Abstract: In this note, we propose and analyse a method for handling interfaces between non-matching grids based on an approach suggested by Nitsche (1971) for the approximation of Dirichlet boundary conditions. The exposition is limited to self-adjoint elliptic problems, using Poisson's equation as a model. A priori and a posteriori error estimates are given. Some numerical results are included.