Showing papers on "Numerical analysis published in 2001"

Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method

Abstract: We describe new algorithms of the locally optimal block preconditioned conjugate gradient (LOBPCG) method for symmetric eigenvalue problems, based on a local optimization of a three-term recurrence, and suggest several other new methods. To be able to compare numerically different methods in the class, with different preconditioners, we propose a common system of model tests, using random preconditioners and initial guesses. As the "ideal" control algorithm, we advocate the standard preconditioned conjugate gradient method for finding an eigenvector as an element of the null-space of the corresponding homogeneous system of linear equations under the assumption that the eigenvalue is known. We recommend that every new preconditioned eigensolver be compared with this "ideal" algorithm on our model test problems in terms of the speed of convergence, costs of every iteration, and memory requirements. We provide such comparison for our LOBPCG method. Numerical results establish that our algorithm is practically as efficient as the ``ideal'' algorithm when the same preconditioner is used in both methods. We also show numerically that the LOBPCG method provides approximations to first eigenpairs of about the same quality as those by the much more expensive global optimization method on the same generalized block Krylov subspace. We propose a new version of block Davidson's method as a generalization of the LOBPCG method. Finally, direct numerical comparisons with the Jacobi--Davidson method show that our method is more robust and converges almost two times faster.

Galerkin proper orthogonal decomposition methods for parabolic problems

Abstract: In this work error estimates for Galerkin proper orthogonal decomposition (POD) methods for linear and certain non-linear parabolic systems are proved. The resulting error bounds depend on the number of POD basis functions and on the time discretization. Numerical examples are included.

Applied Shape Optimization for Fluids

Abstract: Introduction 1. Optimal Shape Design 2. Partial Differential Equations for Fluids 3. Some Numerical Methods for Fluids and Examples 4. Automatic Differentiation 5. Optimization Platform and Implementation Issues 6. Consistent Approximations and Approximate Gradients 7. Numerical Results on Shape Optimization 8. Numerical Results on Shape Optimization for Unsteady Flows Index

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.

Theoretical Numerical Analysis: A Functional Analysis Framework

Abstract: Preface 1 Linear Spaces 2 Linear Operators on Normed Spaces 3 Approximation Theory 4 Nonlinear Equations and Their Solution by Iteration 5 Finite Difference Method 6 Sobolev Spaces 7 Variational Formulations of Elliptic Boundary Value Problems 8 The Galerkin Method and Its Variants 9 Finite Element Analysis 10 Elliptic Variational Inequalities and Their Numerical Approximations 11 Numerical Solution of Fredholm Integral Equations of the Second Kind 12 Boundary Integral Equations References Index.

The Method of Regularized Stokeslets

Abstract: A numerical method for computing Stokes flows in the presence of immersed boundaries and obstacles is presented. The method is based on the smoothing of the forces, leading to regularized Stokeslets. The resulting expressions provide the pressure and velocity field as functions of the forcing. The latter expression can also be inverted to find the forces that impose a given velocity boundary condition. The numerical examples presented demonstrate the wide applicability of the method and its properties. Solutions converge with second-order accuracy when forces are exerted along smooth boundaries. Examples of segmented boundaries and forcing at random points are also presented.

Discontinuous hp -Finite Element Methods for Advection-Diffusion-Reaction Problems

Abstract: We consider the hp-version of the discontinuous Galerkin finite element method (DGFEM) for second-order partial differential equations with nonnegative characteristic form. This class of equations includes second-order elliptic and parabolic equations, advection-reaction equations, as well as problems of mixed hyperbolic-elliptic-parabolic type. Our main concern is the error analysis of the method in the absence of streamline-diffusion stabilization. In the hyperbolic case, an hp-optimal error bound is derived; here, we consider only advection-reaction problems which satisfy a certain (standard) positivity condition. In the self-adjoint elliptic case, an error bound that is h-optimal and p-suboptimal by $\frac{1}{2}$ a power of p is obtained. These estimates are then combined to deduce an error bound in the general case. For elementwise analytic solutions the method exhibits exponential rates of convergence under p-refinement. The theoretical results are illustrated by numerical experiments.

High order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD

Abstract: In recent years high order numerical methods have been widely used in computational fluid dynamics (CFD), to effectively resolve complex flow features using meshes which are reasonable for today''s computers. In this paper we review and compare three types of high order methods being used in CFD, namely the weighted essentially non-oscillatory (WENO) finite difference methods, the WENO finite volume methods, and the discontinuous Galerkin (DG) finite element methods. We summarize the main features of these methods, from a practical user''s point of view, indicate their applicability and relative strength, and show a few selected numerical examples to demonstrate their performance on illustrative model CFD problems.

A computational scheme for linear and non-linear composites with arbitrary phase contrast

Abstract: A numerical method making use of fast Fourier transforms has been proposed in Moulinec and Suquet (1994, 1998) to investigate the effective properties of linear and non-linear composites. This method is based on an iterative scheme the rate of convergence of which is proportional to the contrast between the phases. Composites with high contrast (typically above 104) or infinite contrast (those containing voids or rigid inclusions or highly non-linear materials) cannot be handled by the method. This paper presents two modified schemes. The first one is an accelerated scheme for composites with high contrast which extends to elasticity a scheme initially proposed in Eyre and Milton (1999). Its rate of convergence varies as the square root of the contrast. The second scheme, adequate for composites with infinite contrast, is based on an augmented Lagrangian method. The resulting saddle-point problem involves three steps. The first step consists of solving a linear elastic problem, using the fast Fourier transform method. In the second step, a non-linear problem is solved at each individual point in the volume element. The third step consists of updating the Lagrange multiplier. Applications of this scheme to rigidly reinforced and to voided composites are shown. Copyright © 2001 John Wiley & Sons, Ltd.

A finite element pressure gradient stabilization¶for the Stokes equations based on local projections

Abstract: We present a variant of the classical weighted least-squares stabilization for the Stokes equations. Compared to the original formulation, the new method has improved accuracy properties, especially near boundaries. Furthermore, no modification of the right-hand side is needed, and implementation is simplified, especially for generalizations to more complicated equations. The approach is based on local projections of the residual terms which are used in order to achieve consistency of the method, avoiding local evaluation of the strong form of the differential operator. We prove stability and give a priori and a posteriori error estimates. We show convergence of an iterative method which uses a simplified stabilized discretization as preconditioner. Numerical experiments indicate that the approach presented is at least as accurate as the original method, but offers some algorithmic advantages. The ideas presented here also apply to the Navier–Stokes equations. This is the topic of forthcoming work.

A kinetic scheme for the Saint-Venant system¶with a source term

Abstract: The aim of this paper is to present a numerical scheme to compute Saint-Venant equations with a source term, due to the bottom topography, in a one-dimensional framework, which satisfies the following theoretical properties: it preserves the steady state of still water, satisfies an entropy inequality, preserves the non-negativity of the height of water and remains stable with a discontinuous bottom This is achieved by means of a kinetic approach to the system, which is the departing point of the method developed here In this context, we use a natural description of the microscopic behavior of the system to define numerical fluxes at the interfaces of an unstructured mesh We also use the concept of cell-centered conservative quantities (as usual in the finite volume method) and upwind interfacial sources as advocated by several authors We show, analytically and also by means of numerical results, that the above properties are satisfied

A local discontinuous Galerkin method for KdV-type equations

Abstract: In this paper we develop a local discontinuous Galerkin method for solving KdV type equations containing third derivative terms in one and two space dimensions. The method is based on the framework of the discontinuous Galerkin method for conservation laws and the local discontinuous Galerkin method for viscous equations containing second derivatives; however, the guiding principle for intercell fluxes and nonlinear stability is new. We prove L2 stability and a cell entropy inequality for the square entropy for a class of nonlinear PDEs of this type in both one and multiple space dimensions, and we give an error estimate for the linear cases in the one-dimensional case. The stability result holds in the limit case when the coefficients to the third derivative terms vanish; hence the method is especially suitable for problems which are "convection dominated," i.e., those with small second and third derivative terms. Numerical examples are shown to illustrate the capability of this method. The method has the usual advantage of local discontinuous Galerkin methods, namely, it is extremely local and hence efficient for parallel implementations and easy for h-p adaptivity.

An approach to simulate the motion of spherical and non-spherical fuel particles in combustion chambers

Abstract: The objective of this paper is to identify a numerical method to simulate motion of a packed or fluidized bed of fuel particles in combustion chambers, such as a grate furnace and a rotary kiln. Therefore, the various numerical methods applied in the areas of granular matter and molecular dynamics were reviewed extensively. As a result, a time driven approach was found to be suited for the numerical simulation of particle motion in combustion chambers. Furthermore, this method can also be employed to moving boundaries which are required for the present application e.g. travelling grate. The method works in a Lagrangian frame of reference, which uses the position and orientation of particles as independent variables. These are obtained by time integration of the three-dimensional dynamics equations derived from the classical Newtonian approach for each particle. This includes the keeping track of all forces and momentums acting on each particle at every time step. Viscoelastic contact forces include normal and tangential components with viscoelastic models for energy dissipation and friction. The particle shapes are approximated by spheres and ellipsoids with a varying size and ratio of the semi-axis accounting for the variety of particle geometries in a combustion chamber. For these shapes the overlap of particles during contact is expressed by a polynomial of 4th order in the two-dimensional case and a polynomial of 6th order in the three-dimensional case. A new algorithm to detect two-dimensional elliptical particle contact with sufficient accuracy was developed. It is based on a sequence of coordinate transformations and has demonstrated its reliability in numerous applications. Finally, the method was applied to simulate the motion of spherical and elliptical particles in a rectangular enclosure, on a travelling grate, and in a rotary kiln.

On the Accuracy of the Finite Volume Element Method Based on Piecewise Linear Polynomials

Abstract: We present a general error estimation framework for a finite volume element (FVE) method based on linear polynomials for solving second-order elliptic boundary value problems. This framework treats the FVE method as a perturbation of the Galerkin finite element method and reveals that regularities in both the exact solution and the source term can affect the accuracy of FVE methods. In particular, the error estimates and counterexamples in this paper will confirm that the FVE method cannot have the standard O(h2) convergence rate in the L2 norm when the source term has the minimum regularity, only being in L2, even if the exact solution is in H2.

Numerical simulation methods for rough surface scattering

Abstract: Numerical methods are of great importance in the study of electromagnetic scattering from random rough surfaces. This review provides an overview of rough surface scattering and application areas of current interest, and surveys research in numerical simulation methods for both one- and two-dimensional surfaces. Approaches considered include numerical methods based on analytical scattering approximations, differential equation methods and surface integral equation methods. Emphasis is placed on recent advances such as rapidly converging iterative solvers for rough surface problems and fast methods for increasing the computational efficiency of integral equation solvers.

Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth

Abstract: In this paper we examine spatio-temporal pattern formation in reaction-diffusion systems on the surface of the unit sphere in 3D. We first generalise the usual linear stability analysis for a two-chemical system to this geometrical context. Noting the limitations of this approach (in terms of rigorous prediction of spatially heterogeneous steady-states) leads us to develop, as an alternative, a novel numerical method which can be applied to systems of any dimension with any reaction kinetics. This numerical method is based on the method of lines with spherical harmonics and uses fast Fourier transforms to expedite the computation of the reaction kinetics. Numerical experiments show that this method efficiently computes the evolution of spatial patterns and yields numerical results which coincide with those predicted by linear stability analysis when the latter is known. Using these tools, we then investigate the role that pre-pattern (Turing) theory may play in the growth and development of solid tumours. The theoretical steady-state distributions of two chemicals (one a growth activating factor, the other a growth inhibitory factor) are compared with the experimentally and clinically observed spatial heterogeneity of cancer cells in small, solid spherical tumours such as multicell spheroids and carcinomas. Moreover, we suggest a number of chemicals which are known to be produced by tumour cells (autocrine growth factors), and are also known to interact with one another, as possible growth promoting and growth inhibiting factors respectively. In order to connect more concretely the numerical method to this application, we compute spatially heterogeneous patterns on the surface of a growing spherical tumour, modelled as a moving-boundary problem. The numerical results strongly support the theoretical expectations in this case. Finally in an appendix we give a brief analysis of the numerical method.

New anisotropic a priori error estimates

Abstract: We prove a priori anisotropic estimates for the $L^2$ and $H^1$ interpolation error on linear finite elements. The full information about the mapping from a reference element is employed to separate the contribution to the elemental error coming from different directions. This new $H^1$ error estimate does not require the “maximal angle condition”. The analysis has been carried out for the 2D case, but may be extended to three dimensions. Numerical experiments have been carried out to test our theoretical results.

The numerical solution of fractional differential equations: Speed versus accuracy

Abstract: This paper is concerned with the development of efficient algorithms for the approximate solution of fractional differential equations of the form Dαy(t)=f(t,y(t)), α∈R+−N.(†)

Grid Convergence Error Analysis for Mixed-Order Numerical Schemes

Abstract: New developments are presented in the area of grid convergence error analysis and error estimation for mixed-order numerical schemes. A mixed-order scheme is defined here as a numerical method where the order of the local truncation error varies either spatially (e.g., at a shock wave) or for different terms in the governing equations (e.g., first-order convection with second-order diffusion). The case examined herein is the Mach 8 laminar flow of a perfect gas over a sphere-cone geometry. This flowfield contains a strong bow shock wave where the formally second-order numerical scheme is reduced to first order via a flux limiting procedure. The mixedorder error analysis method allows for non-monotone behavior in the solutions variables as the mesh is refined. Non-monotonicity in the local solution variables is shown to arise from a cancellation of first- and second-order error terms for the present case. The proposed error estimator, which is based on the mixed-order analysis, is shown to provide good estimates of the actual error. Furthermore, this error estimator nearly always provides conservative error estimates, in the sense that the actual error is less than the error estimate, for the case examined.

Calculation of Wing Flutter by a Coupled Fluid-Structure Method

Abstract: An integrated computational fluid dynamics (CFD) and computational structural dynamics (CSD) method is developed for the simulation and prediction of flutter. The CFD solver is based on an unsteady, parallel, multiblock, multigrid finite volume algorithm for the Euler/Navier-Stokes equations. The CSD solver is based on the time integration of modal dynamic equations extracted from full finite element analysis. A general multiblock deformation grid method is used to generate dynamically moving grids for the unsteady flow solver. The solutions of the flowfield and the structural dynamics are coupled strongly in time by a fully implicit method. The coupled CFD-CSD method simulates the aeroelastic system directly on the time domain to determine the stability of the aeroelastic system. The unsteady solver with the moving grid algorithm is also used to calculate the harmonic and/or indicial responses of an aeroelastic system, in an uncoupled manner, without solving the structural equations. Flutter boundary is then determined by solving the flutter equation on the frequency domain with the indicial responses as input. Computations are performed for a two-dimensional wing aeroelastic model and the three-dimensional AGARD 445.6 wing. Flutter boundary predictions by both the coupled CFD-CSD method and the indicial method are presented and compared with experimental data for the AGARD 445.6 wing.

Analyzing circuits with widely separated time scales using numerical PDE methods

Abstract: Widely separated time scales arise in many kinds of circuits, e,g., switched-capacitor filters, mixers, switching power converters, etc. Numerical solution of such circuits is often difficult, especially when strong nonlinearities are present. In this paper, the author presents a mathematical formulation and numerical methods for analyzing a broad class of such circuits or systems. The key idea is to use multiple time variables, which enable signals with widely separated rates of variation to be represented efficiently. This results in the transformation of differential equation descriptions of a system to partial differential ones, in effect decoupling different rates of variation from each other. Numerical methods can then be used to solve the partial differential equations (PDEs). In particular, time-domain methods can be used to handle the hitherto difficult case of strong nonlinearities together with widely separated rates of signal variation. The author examines methods for obtaining quasiperiodic and envelope solutions, and describes how the PDE formulation unifies existing techniques for separated-time-constant problems. Several applications are described. Significant computation and memory savings result from using the new numerical techniques, which also scale gracefully with problem size.