Showing papers on "Numerical analysis published in 2010"
TL;DR: A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs).
Abstract: A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem. The original empirical interpolation method (EIM) is a modification of POD that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD. We propose a discrete empirical interpolation method (DEIM), a variant that is suitable for reducing the dimension of systems of ordinary differential equations (ODEs) of a certain type. As presented here, it is applicable to ODEs arising from finite difference discretization of time dependent PDEs and/or parametrically dependent steady state problems. However, the approach extends to arbitrary systems of nonlinear ODEs with minor modification. Our contribution is a greatly simplified description of the EIM in a finite-dimensional setting that possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the one-dimensional FitzHugh-Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captures nonlinear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results.
2,110 citations
Book•
21 Jul 2010TL;DR: This book describes the class of numerical methods based on generalized polynomial chaos (gPC), an extension of the classical spectral methods of high-dimensional random spaces designed to simulate complex systems subject to random inputs.
Abstract: The first graduate-level textbook to focus on fundamental aspects of numerical methods for stochastic computations, this book describes the class of numerical methods based on generalized polynomial chaos (gPC). These fast, efficient, and accurate methods are an extension of the classical spectral methods of high-dimensional random spaces. Designed to simulate complex systems subject to random inputs, these methods are widely used in many areas of computer science and engineering. The book introduces polynomial approximation theory and probability theory; describes the basic theory of gPC methods through numerical examples and rigorous development; details the procedure for converting stochastic equations into deterministic ones; using both the Galerkin and collocation approaches; and discusses the distinct differences and challenges arising from high-dimensional problems. The last section is devoted to the application of gPC methods to critical areas such as inverse problems and data assimilation. Ideal for use by graduate students and researchers both in the classroom and for self-study, Numerical Methods for Stochastic Computations provides the required tools for in-depth research related to stochastic computations.The first graduate-level textbook to focus on the fundamentals of numerical methods for stochastic computations Ideal introduction for graduate courses or self-study Fast, efficient, and accurate numerical methods Polynomial approximation theory and probability theory included Basic gPC methods illustrated through examples
1,583 citations
TL;DR: In this paper, a new robust and efficient approach for modeling discrete cracks in mesh-free methods is described, where the crack is modeled by splitting particles located on opposite sides of the associated crack segments and make use of the visibility method in order to describe the crack kinematics.
874 citations
TL;DR: The proposed algorithm has been successfully tested on two very important evolution equations namely Fitzhugh–Nagumo equation and Sharma–Tasso–Olver equation and results are very encouraging.
474 citations
TL;DR: In this paper, an explicit and easily implementable numerical method for such an SDE was proposed, which converges strongly with the standard order one-half to the exact solution of the SDE.
Abstract: On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other hand, the implicit Euler scheme is known to converge strongly to the exact solution of such an SDE. Implementations of the implicit Euler scheme, however, require additional computational effort. In this article we therefore propose an explicit and easily implementable numerical method for such an SDE and show that this method converges strongly with the standard order one-half to the exact solution of the SDE. Simulations reveal that this explicit strongly convergent numerical scheme is considerably faster than the implicit Euler scheme.
367 citations
TL;DR: A numerical scheme applicable to arbitrarily-structured C-grids is presented for the nonlinear shallow-water equations, using the vector-invariant form of the momentum equation to guarantee that mass, velocity and potential vorticity evolve in a consistent and compatible manner.
357 citations
TL;DR: A high order discontinuous Galerkin method is proposed which can maintain the still water steady state exactly, and at the same time can preserve the non-negativity of the water height without loss of mass conservation.
350 citations
TL;DR: A physically consistent phase-field model that admits an energy law is proposed, and several energy stable, efficient, and accurate time discretization schemes for the coupled nonlinear phase- field model are constructed and analyzed.
Abstract: Modeling and numerical approximation of two-phase incompressible flows with different densities and viscosities are considered. A physically consistent phase-field model that admits an energy law is proposed, and several energy stable, efficient, and accurate time discretization schemes for the coupled nonlinear phase-field model are constructed and analyzed. Ample numerical experiments are carried out to validate the correctness of these schemes and their accuracy for problems with large density and viscosity ratios.
342 citations
TL;DR: This paper develops a fast finite difference method for fractional diffusion equations, which only requires storage of O(N) and computational cost of O (Nlog^2N) while retaining the same accuracy and approximation property as the regular finite Difference method.
337 citations
TL;DR: A new unified family of arbitrary high order accurate explicit one-step finite volume and discontinuous Galerkin schemes on unstructured triangular and tetrahedral meshes for the solution of the compressible Navier–Stokes equations is proposed.
TL;DR: The Haar wavelet operational matrix is derived and used to solve the fractional order differential equations including the Bagley-Torvik, Ricatti and composite fractional oscillation equations.
TL;DR: A new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes, and without resolving the interfaces is introduced.
Abstract: We introduce a new multiscale finite element method which is
able to accurately capture solutions of elliptic interface problems with high
contrast coefficients by using only coarse quasiuniform meshes, and without
resolving the interfaces. A typical application would be the modelling of flow
in a porous medium containing a number of inclusions of low (or high) permeability
embedded in a matrix of high (respectively low) permeability. Our
method is H^1- conforming, with degrees of freedom at the nodes of a triangular
mesh and requiring the solution of subgrid problems for the basis functions on
elements which straddle the coefficient interface but which use standard linear
approximation otherwise. A key point is the introduction of novel coefficientdependent
boundary conditions for the subgrid problems. Under moderate
assumptions, we prove that our methods have (optimal) convergence rate of
O(h) in the energy norm and O(h^2) in the L_2 norm where h is the (coarse)
mesh diameter and the hidden constants in these estimates are independent
of the “contrast” (i.e. ratio of largest to smallest value) of the PDE coefficient.
For standard elements the best estimate in the energy norm would be
O(h^(1/2−e)) with a hidden constant which in general depends on the contrast.
The new interior boundary conditions depend not only on the contrast of the
coefficients, but also on the angles of intersection of the interface with the
element edges.
TL;DR: An edge-based smoothed finite element method (ES-FEM) for static, free vibration and buckling analyses of Reissner-Mindlin plates using 3-node triangular elements is studied in this paper.
TL;DR: A combined form of the Laplace transform method with the Adomian decomposition method is developed for analytic treatment of the nonlinear Volterra integro-differential equations.
Book•
14 Sep 2010
TL;DR: In this article, a finite-difference approximation of the Wave Equation is presented. But this approach is not suitable for the case of large numbers of fast waves and nonreflecting boundary conditions.
Abstract: Introduction*Ordinary Differential Equations*Finite-Difference Approximation of the Wave Equation*Diffusion, Sources and Sinks*Series Expansion Methods*Finite-Volume Methods*Semi-Lagrangian Methods*Physically Insignificant Fast Waves*Nonreflecting Boundary Conditions*Appendix
TL;DR: A reliable new algorithm of DTM is proposed, namely multi-step DTM, which will increase the interval of convergence for the series solution, and is applied to Lotka-Volterra, Chen and Lorenz systems.
Abstract: The differential transform method (DTM) is an analytical and numerical method for solving a wide variety of differential equations and usually gets the solution in a series form. In this paper, we propose a reliable new algorithm of DTM, namely multi-step DTM, which will increase the interval of convergence for the series solution. The multi-step DTM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions for systems of differential equations. This new algorithm is applied to Lotka-Volterra, Chen and Lorenz systems. Then, a comparative study between the new algorithm, multi-step DTM, classical DTM and the classical Runge-Kutta method is presented. The results demonstrate reliability and efficiency of the algorithm developed.
TL;DR: Numerical solutions using finite element methods are considered for transient flow in a porous medium coupled to free flow in embedded conduits, for example, for groundwater flows in karst aquifers.
Abstract: Numerical solutions using finite element methods are considered for transient flow in a porous medium coupled to free flow in embedded conduits. Such situations arise, for example, for groundwater flows in karst aquifers. The coupled flow is modeled by the Darcy equation in a porous medium and the Stokes equations in the conduit domain. On the interface between the matrix and conduit, Beavers-Joseph interface conditions, instead of the simplified Beavers-Joseph-Saffman conditions, are imposed. Convergence and error estimates for finite element approximations are obtained. Numerical experiments illustrate the validity of the theoretical results.
TL;DR: In this article, a unified finite element approach to fully coupled cardiac electromechanics is proposed, where the intrinsic coupling arises from both the excitation-induced contraction of cardiac cells and the deformation-induced generation of current due to the opening of ion channels.
Abstract: This manuscript is concerned with a novel, unified finite element approach to fully coupled cardiac electromechanics The intrinsic coupling arises from both the excitation-induced contraction of cardiac cells and the deformation-induced generation of current due to the opening of ion channels In contrast to the existing numerical approaches suggested in the literature, which devise staggered algorithms through distinct numerical methods for the respective electrical and mechanical problems, we propose a fully implicit, entirely finite element-based modular approach To this end, the governing differential equations that are coupled through constitutive equations are recast into the corresponding weak forms through the conventional isoparametric Galerkin method The resultant non-linear weighted residual terms are then consistently linearized The system of coupled algebraic equations obtained through discretization is solved monolithically The put-forward modular algorithmic setting leads to an unconditionally stable and geometrically flexible framework that lays a firm foundation for the extension of constitutive equations towards more complex ionic models of cardiac electrophysiology and the strain energy functions of cardiac mechanics The performance of the proposed approach is demonstrated through three-dimensional illustrative initial boundary-value problems that include a coupled electromechanical analysis of a biventricular generic heart model
TL;DR: In this paper, the displacement discontinuity across crack surface is modeled by independent cover functions over different physical covers, while additional functions, extracted from the asymptotic near tip field, are incorporated into cover functions of singular physical covers to reflect the stress singularity around the crack tips.
Abstract: The numerical manifold method is a cover-based method using mathematical covers that are independent of the physical domain. As the unknowns are defined on individual physical covers, the numerical manifold method is very suitable for modeling discontinuities. This paper focuses on modeling complex crack propagation problems containing multiple or branched cracks. The displacement discontinuity across crack surface is modeled by independent cover functions over different physical covers, while additional functions, extracted from the asymptotic near tip field, are incorporated into cover functions of singular physical covers to reflect the stress singularity around the crack tips. In evaluating the element matrices, Gaussian quadrature is used over the sub-triangles of the element, replacing the simplex integration over the whole element. First, the method is validated by evaluating the fracture parameters in two examples involving stationary cracks. The results show good agreement with the reference solutions available. Next, three crack propagation problems involving multiple and branched cracks are simulated. It is found that when the crack growth increment is taken to be 0.5h≤da≤0.75h, the crack growth paths converge consistently and are satisfactory.
TL;DR: The method of harmonic linearization, numerical methods, and the applied bifurcation theory together discover new opportunities for analysis of periodic oscillations of control systems and application of this technique for attractor localization of generalized Chua's systems is given.
TL;DR: An alternative approach of the variational iteration method is presented, then the convergence of the method for nonlinear differential equations is studied to address the sufficient condition for convergence and the error estimate.
TL;DR: It is proved that the variational solution of the SSFPDE exists and is unique, and a fully discrete approximating system is obtained, which has a unique solution according to the Lax–Milgram theorem.
TL;DR: This method improves the accuracy of split-step finite difference method by introducing a compact scheme for discretization of space variable while this improvement does not reduce the stability range and does not increase the computational cost.
Book•
26 Apr 2010
TL;DR: Magnets for Accelerators Algebraic Structures and Vector Fields Classical Vector Analysis Maxwell's Equations and Boundary Value Problems Fields and Potentials of Line Currents Field Harmonics IronDominated Magnets Coil-DominatedMagnets Complex Analysis Methods for Magnet Design Field Diffusion Elementary Beam Optics and Field Requirements Reference Frames and Magnet Polarities Finite-element Formulations Discretization Coupling of Boundary and Finite Elements Superconductor Magnetization Interstrand Coupling Currents Quench Simulation Differential Geometry Applied to Coil-End Design Mathematical Optimization Techniques
Abstract: Magnets for Accelerators Algebraic Structures and Vector Fields Classical Vector Analysis Maxwell's Equations and Boundary Value Problems Fields and Potentials of Line Currents Field Harmonics Iron-Dominated Magnets Coil-Dominated Magnets Complex Analysis Methods for Magnet Design Field Diffusion Elementary Beam Optics and Field Requirements Reference Frames and Magnet Polarities Finite-Element Formulations Discretization Coupling of Boundary and Finite Elements Superconductor Magnetization Interstrand Coupling Currents Quench Simulation Differential Geometry Applied to Coil-End Design Mathematical Optimization Techniques Material Property Data for Quench Simulations
TL;DR: It is shown that the ADFEM (and the AFEM on nonconforming meshes) yields a decay rate of energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.
Abstract: We analyze an adaptive discontinuous finite element method (ADFEM) for symmetric second order linear elliptic operators. The method is formulated on nonconforming meshes made of simplices or quadrilaterals, with any polynomial degree and in any dimension $\geq2$. We prove that the ADFEM is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops. We design a refinement procedure that maintains the level of nonconformity uniformly bounded and prove that the approximation classes using continuous and discontinuous finite elements are equivalent. The geometric decay and the equivalence of classes are instrumental in deriving the optimal cardinality of the ADFEM. We show that the ADFEM (and the AFEM on nonconforming meshes) yields a decay rate of energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.
Book•
2 Dec 2010
TL;DR: The text most notably provides a detailed exposition of the Sinc separation of variables method for numerically solving the full range of partial differential equations (PDEs) of interest to scientists and engineers.
Abstract: Handbook of Sinc Numerical Methods presents an ideal road map for handling general numeric problems Reflecting the authors advances with Sinc since 1995, the text most notably provides a detailed exposition of the Sinc separation of variables method for numerically solving the full range of partial differential equations (PDEs) of interest to scientists and engineers This new theory, which combines Sinc convolution with the boundary integral equation (IE) approach, makes for exponentially faster convergence to solutions of differential equations The basis for the approach is the Sinc method of approximating almost every type of operation stemming from calculus via easily computed matrices of very low dimension The CD-ROM of this handbook contains roughly 450 MATLAB programs corresponding to exponentially convergent numerical algorithms for solving nearly every computational problem of science and engineering While the book makes Sinc methods accessible to users wanting to bypass the complete theory, it also offers sufficient theoretical details for readers who do want a full working understanding of this exciting area of numerical analysis
TL;DR: A computational approach that can be viewed as a continuous-time updating procedure that works with the increments rather than the full tensor and avoids the computation of decompositions of large matrices is studied.
Abstract: For the approximation of time-dependent data tensors and of solutions to tensor differential equations by tensors of low Tucker rank, we study a computational approach that can be viewed as a continuous-time updating procedure. This approach works with the increments rather than the full tensor and avoids the computation of decompositions of large matrices. In this method, the derivative is projected onto the tangent space of the manifold of tensors of Tucker rank $(r_1,\dots,r_N)$ at the current approximation. This yields nonlinear differential equations for the factors in a Tucker decomposition, suitable for numerical integration. Approximation properties of this approach are analyzed.
TL;DR: In this article, a node-based smoothed finite element method (NS-FEM) was proposed for the solid mechanics problems, which is further extended to more complicated visco-elastoplastic analyses of 2D and 3D solids using triangular and tetrahedral meshes.
TL;DR: In this paper, accurate finite element methods for solving 3-D Poisson-Nernst-Planck (PNP) equations with singular permanent charges for electrodiffusion in solvated biomolecular systems are developed and shown to be accurate and stable by various test problems.