Showing papers on "Numerical analysis published in 2006"
Book•
01 Jan 2006
TL;DR: This work considers the semi-Lagrangian discontinuous Galerkin method for the Vlasov-Poisson system and discusses the performance of this method and compares it to cubic spline interpolation, where appropriate.
Abstract: The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential
equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods.
1,456 citations
TL;DR: This paper provides theoretical analysis for a cubic regularization of Newton method as applied to unconstrained minimization problem and proves general local convergence results for this scheme.
Abstract: In this paper, we provide theoretical analysis for a cubic regularization of Newton method as applied to unconstrained minimization problem. For this scheme, we prove general local convergence results. However, the main contribution of the paper is related to global worst-case complexity bounds for different problem classes including some nonconvex cases. It is shown that the search direction can be computed by standard linear algebra technique.
960 citations
TL;DR: In this article, the effect of finite thermal conductivity and recirculation in droplets can be taken into account using the so-called parabolic model, which is a reasonable compromise between accuracy and CPU efficiency.
724 citations
TL;DR: It is shown that the fractional Crank-Nicholson method based on the shifted Grunwald formula is unconditionally stable and compared with the exact analytical solution for its order of convergence.
557 citations
TL;DR: In this paper, homotopy perturbation method (HPM), which does not need small parameters in the equations, is compared with the perturbations and numerical methods in the heat transfer field.
496 citations
TL;DR: The development of the highly accurate ADER–DG approach for tetrahedral meshes provides a numerical technique to approach 3-D wave propagation problems in complex geometry with unforeseen accuracy.
Abstract: SUMMARY
We present a new numerical method to solve the heterogeneous elastic wave equations formulated as a linear hyperbolic system using first-order derivatives with arbitrary high-order accuracy in space and time on 3-D unstructured tetrahedral meshes. The method combines the Discontinuous Galerkin (DG) Finite Element (FE) method with the ADER approach using Arbitrary high-order DERivatives for flux calculation. In the DG framework, in contrast to classical FE methods, the numerical solution is approximated by piecewise polynomials which allow for discontinuities at element interfaces. Therefore, the well-established theory of numerical fluxes across element interfaces obtained by the solution of Riemann-Problems can be applied as in the finite volume framework. To define a suitable flux over the element surfaces, we solve so-called Generalized Riemann-Problems (GRP) at the element interfaces. The GRP solution provides simultaneously a numerical flux function as well as a time-integration method. The main idea is a Taylor expansion in time in which all time-derivatives are replaced by space derivatives using the so-called Cauchy–Kovalewski or Lax–Wendroff procedure which makes extensive use of the governing PDE. The numerical solution can thus be advanced for one time step without intermediate stages as typical, for example, for classical Runge–Kutta time stepping schemes. Due to the ADER time-integration technique, the same approximation order in space and time is achieved automatically. Furthermore, the projection of the tetrahedral elements in physical space on to a canonical reference tetrahedron allows for an efficient implementation, as many computations of 3-D integrals can be carried out analytically beforehand. Based on a numerical convergence analysis, we demonstrate that the new schemes provide very high order accuracy even on unstructured tetrahedral meshes and computational cost and storage space for a desired accuracy can be reduced by higher-order schemes. Moreover, due to the choice of the basis functions for the piecewise polynomial approximation, the new ADER–DG method shows spectral convergence on tetrahedral meshes. An application of the new method to a well-acknowledged test case and comparisons with analytical and reference solutions, obtained by different well-established methods, confirm the performance of the proposed method. Therefore, the development of the highly accurate ADER–DG approach for tetrahedral meshes provides a numerical technique to approach 3-D wave propagation problems in complex geometry with unforeseen accuracy.
433 citations
TL;DR: In this paper, a numerical method is proposed to analyse the electromagnetic behavior of systems including high-temperature superconductors (HTSCs) in time-varying external fields and superconducting cables carrying AC transport current.
Abstract: A numerical method is proposed to analyse the electromagnetic behaviour of systems including high-temperature superconductors (HTSCs) in time-varying external fields and superconducting cables carrying AC transport current. The E–J constitutive law together with an H-formulation is used to calculate the current distribution and electromagnetic fields in HTSCs, and the magnetization of HTSCs; then the forces in the interaction between the electromagnet and the superconductor and the AC loss of the superconducting cable can be obtained. This numerical method is based on solving the partial differential equations time dependently and is adapted to the commercial finite element software Comsol Multiphysics 3.2. The advantage of this method is to make the modelling of the superconductivity simple, flexible and extendable.
428 citations
TL;DR: In this paper, a new family of stabilized methods for the Stokes problem is proposed, which are parameter free, do not require calculation of higher order derivatives or edge-based data structures, and always lead to symmetric linear systems.
Abstract: We present a new family of stabilized methods for the Stokes problem. The focus of the paper is on the lowest order velocity-pressure pairs. While not LBB compliant, their simplicity and attractive computational properties make these pairs a popular choice in engineering practice. Our stabilization approach is motivated by terms that characterize the LBB "deficiency" of the unstable spaces. The stabilized methods are defined by using these terms to modify the saddle-point Lagrangian associated with the Stokes equations. The new stabilized methods offer a number of attractive computational properties. In contrast to other stabilization procedures, they are parameter free, do not require calculation of higher order derivatives or edge-based data structures, and always lead to symmetric linear systems. Furthermore, the new methods are unconditionally stable, achieve optimal accuracy with respect to solution regularity, and have simple and straightforward implementations. We present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods.
367 citations
TL;DR: A class of finite difference methods for solving fractional diffusion equations is considered, an extension of the weighted average methods for ordinary (non-fractional) diffusion equations, and a simple and accurate stability criterion valid for different discretization schemes of the fractional derivative, arbitrary weight factor, and arbitrary order is found.
333 citations
TL;DR: In this paper, the mathematical properties of the inverse transformation were discussed and six numerical methods for performing the inversion were presented, including a Newton-Raphson scheme and a guess from the previous time step.
Abstract: Conservative numerical schemes for general relativistic magnetohydrodynamics (GRMHD) require a method for transforming between "conserved" variables such as momentum and energy density and "primitive" variables such as rest-mass density, internal energy, and components of the four-velocity. The forward transformation (primitive to conserved) has a closed-form solution, but the inverse transformation (conserved to primitive) requires the solution of a set of five nonlinear equations. Here we discuss the mathematical properties of the inverse transformation and present six numerical methods for performing the inversion. The first method solves the full set of five nonlinear equations directly using a Newton-Raphson scheme and a guess from the previous time step. The other methods reduce the five nonlinear equations to either one or two nonlinear equations that are solved numerically. Comparisons between the methods are made using a survey over phase space, a two-dimensional explosion problem, and a general relativistic MHD accretion disk simulation. The run time of the methods is also examined. Code implementing the schemes is available with the electronic edition of the article.
291 citations
TL;DR: A fully derivative-free spectral residual method for solving largescale nonlinear systems of equations that uses in a systematic way the residual vector as a search direction, a spectral steplength that produces a nonmonotone process and a globalization strategy that allows for this nonmonothone behavior.
Abstract: A fully derivative-free spectral residual method for solving largescale nonlinear systems of equations is presented. It uses in a systematic way the residual vector as a search direction, a spectral steplength that produces a nonmonotone process and a globalization strategy that allows for this nonmonotone behavior. The global convergence analysis of the combined scheme is presented. An extensive set of numerical experiments that indicate that the new combination is competitive and frequently better than well-known Newton-Krylov methods for largescale problems is also presented.
TL;DR: This paper investigates the computational aspects of the Galerkin approximation using continuous piecewise polynomial basis functions on a regular triangulation of the domain and demonstrates approximations to FADEs.
TL;DR: Second-order error bounds of the time discretization and regularity estimates for the solution are shown in a unified way under weak assumptions on the data in a Banach space framework.
Abstract: We propose and study a numerical method for time discretization of linear and semilinear integro-partial differential equations that are intermediate between diffusion and wave equations, or are subdiffusive. The method uses convolution quadrature based on the second-order backward differentiation formula. Second-order error bounds of the time discretization and regularity estimates for the solution are shown in a unified way under weak assumptions on the data in a Banach space framework. Numerical experiments illustrate the theoretical results.
TL;DR: An algebraic representation that is useful for matrices with off-diagonal blocks of low numerical rank and a fast and stable solver for linear systems of equations in which the coefficient matrix has this representation is presented.
Abstract: We consider an algebraic representation that is useful for matrices with off-diagonal blocks of low numerical rank. A fast and stable solver for linear systems of equations in which the coefficient matrix has this representation is presented. We also present a fast algorithm to construct the hierarchically semiseparable representation in the general case.
TL;DR: In this paper, a numerical method for form-finding of tensegrity structures is presented, where Eigenvalue analysis and spectral decomposition are carried out iteratively to find the feasible set of force densities that satisfies the requirement on rank deficiency of the equilibrium matrix with respect to the nodal coordinates.
TL;DR: In this paper, an optimal design theory for structures implemented with tuned mass dampers (TMDs) is proposed, and the optimal design parameters of TMDs in terms of damping coefficients and spring constants corresponding to each TMD are determined through minimizing a performance index of structural responses defined in the frequency domain.
TL;DR: The recently introduced local projection stabilization is proposed to apply to the numerical computation of the Oseen equation at high Reynolds number and a priori error estimation techniques are used to prove the convergence of the method.
Abstract: We propose to apply the recently introduced local projection stabilization to the numerical computation of the Oseen equation at high Reynolds number. The discretization is done by nested finite element spaces. Using a priori error estimation techniques, we prove the convergence of the method. The a priori estimates are independent of the local Peclet number and give a sufficient condition for the size of the stabilization parameters in order to ensure optimality of the approximation when the exact solution is smooth. Moreover, we show how this method may be cast in the framework of variational multiscale methods. We indicate what modeling assumptions must be made to use the method for large eddy simulations.
Book•
04 Jan 2006
TL;DR: In this paper, the authors compared the methods of Galerkin and Bogoliubov for steady-state vibrations with those of Krylov-Bogolov and Ritz.
Abstract: Fundamental concepts frequency response of linear systems general response properties matrix analysis natural frequencies and mode shapes singular and defective matrices numerical methods for modal analysis response functions application of response functions - Fourier transforms discrete response calculations systems with symmetric matrices - Langrange's equations continuous systems - Rayleigh's method parametric and nonlinear effects - methods for finding the forced response of weakly-nonlinear systems - Galerkin's method, Ritz's method, Krylov and Bogoliubov's method, comparison between the methods of Galerkin and Krylov-Bogoliubov for steady-state vibrations. Appendices: logical flow diagrams - upper Hessenberg form of a real unsymmetric matrix A(N,N) using Gaussian elimination with interchanges, one iteration of the QR transform, eigenvalues of a real matrix A(N,N) by using the QR transform of the previous appendix, determinant of an upper-Hessenberg matrix by Hyman's method, eigenvectors of a real matrix A(N,N) whose eigenvalues are known, inverse of a complex matrix, one Ruge-Kutta 4th order step.
TL;DR: A survey of variational, symplectic and multi-symplectic discretization methods for Hamiltonian partial differential equations can be found in this paper, where the derivation of methods as well as some of their fundamental geometric properties are discussed.
Abstract: The paper provides an introduction and survey of conservative discretization methods for Hamiltonian partial differential equations. The emphasis is on variational, symplectic and multi-symplectic methods. The derivation of methods as well as some of their fundamental geometric properties are discussed. Basic principles are illustrated by means of examples from wave and fluid dynamics.
Book•
26 Oct 2006
TL;DR: This paper presents the results of a two-week study of the simulation of Evolution Equations and Vector Calculus Fundamentals in COMSOL Multiphysics with MATLAB using the Fokker-Planck Equation Crevice method.
Abstract: Introduction to COMSOL Multiphysics COMSOL Multiphysics and the Basics of Numerical Analysis Analyzing Evolution Equations by the Finite Element Method Multiphysics Extended Multiphysics Nonlinear Dynamics and Linear System Analysis Changing Geometry: Continuation and Moving Boundaries Coupling Variables Revisited: Inverse Problems, Line Integrals, Integral Equations, and Integro-Differential Equations Modeling of Multi-Phase Flow Using the Level Set Method Modeling of Free Surface Flow Problems with Phase Change -- Three Phase Flows Newtonian Flow in Grooved Microchannels Electrokinetic Flow Plasma Simulations via the Fokker-Planck Equation Crevice Corrosion of Steel Under a Disbonded Coating Numerical Simulation of a Magnetohydrodynamic DC Microdevice Vector Calculus Fundamentals in COMSOL Multiphysics with MATLAB.
TL;DR: A mixed finite element method for single phase flow in porous media that reduces to cell-centered finite differences on quadrilateral and simplicial grids and performs well for discontinuous full tensor coefficients is developed.
Abstract: We develop a mixed finite element method for single phase flow in porous media that reduces to cell-centered finite differences on quadrilateral and simplicial grids and performs well for discontinuous full tensor coefficients. Motivated by the multipoint flux approximation method where subedge fluxes are introduced, we consider the lowest order Brezzi-Douglas-Marini (BDM) mixed finite element method. A special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell-centered system for the pressures. Theoretical and numerical results indicate second-order convergence for pressures at the cell centers and first-order convergence for subedge fluxes. Second-order convergence for edge fluxes is also observed computationally if the grids are sufficiently regular.
TL;DR: A novel approach to incorporate high order approximations to derivatives into numerical Hamiltonians such that the resulting numerical schemes are formally high order accurate and inherit the fast convergence from the alternating sweeping strategy.
Abstract: We construct high order fast sweeping numerical methods for computing viscosity solutions of static Hamilton---Jacobi equations on rectangular grids. These methods combine high order weighted essentially non-oscillatory (WENO) approximations to derivatives, monotone numerical Hamiltonians and Gauss---Seidel iterations with alternating-direction sweepings. Based on well-developed first order sweeping methods, we design a novel approach to incorporate high order approximations to derivatives into numerical Hamiltonians such that the resulting numerical schemes are formally high order accurate and inherit the fast convergence from the alternating sweeping strategy. Extensive numerical examples verify efficiency, convergence and high order accuracy of the new methods.
TL;DR: In this paper, the authors present a new numerical method for calculating the transfer of ionizing radiation, called C2-ray (conservative, causal ray-tracing method), which is explicitly photon-conserving, so the depletion of ionising photons by bound-free opacity is guaranteed to equal the photoionizations these photons caused.
TL;DR: A homotopy perturbation method, like Adomian's decomposition method, is proposed for solving the non-singular integral equations of the first kind and the results reveal that the proposed method is very effective and simple.
TL;DR: A system of equations is derived that characterizing the numerical solution of fractional integro-differential equations of the type D q y ( t ) by polynomial spline functions.
TL;DR: In this paper, a new numerical method, named as Traction Image method, is proposed to accurately and efficiently implement the traction-free boundary conditions in finite difference simulation in the presence of surface topography.
Abstract: SUMMARY
In this study, we propose a new numerical method, named as Traction Image method, to accurately and efficiently implement the traction-free boundary conditions in finite difference simulation in the presence of surface topography. In this algorithm, the computational domain is discretized by boundary-conforming grids, in which the irregular surface is transformed into a ‘flat’ surface in computational space. Thus, the artefact of staircase approximation to arbitrarily irregular surface can be avoided. Such boundary-conforming gridding is equivalent to a curvilinear coordinate system, in which the first-order partial differential velocity-stress equations are numerically updated by an optimized high-order non-staggered finite difference scheme, that is, DRP/opt MacCormack scheme. To satisfy the free surface boundary conditions, we extend the Stress Image method for planar surface to Traction Image method for arbitrarily irregular surface by antisymmetrically setting the values of normal traction on the grid points above the free surface. This Traction Image method can be efficiently implemented. To validate this new method, we perform numerical tests to several complex models by comparing our results with those computed by other independent accurate methods. Although some of the testing examples have extremely sloped topography, all tested results show an excellent agreement between our results and those from the reference solutions, confirming the validity of our method for modelling seismic waves in the heterogeneous media with arbitrary shape topography. Numerical tests also demonstrate the efficiency of this method. We find about 10 grid points per shortest wavelength is enough to maintain the global accuracy of the simulation. Although the current study is for 2-D P-SV problem, it can be easily extended to 3-D problem.
TL;DR: This article implements a relatively new analytical technique, the Adomian decomposition method, for solving fractional Riccati differential equations, in which the solution takes the form of a convergent series with easily computable components.
TL;DR: Comparisons are made between Adomian’s decomposition method (ADM) and the exact solution and the proposed homotopy perturbation method to solve quadratic Riccati differential equation.
TL;DR: The present numerical method is applied to both the forced motion and fluid-structure interaction problems and is able to solve fully coupled Navier-Stokes and dynamic equations for the moving body without introducing any iteration.
TL;DR: The fuzzy decision maps (FDM), which incorporates the eigenvalue method, the fuzzy cognitive maps (FCM), and the weighting equation, to overcome the problem of preferential independent and the shortcomings of the ANP.