Showing papers on "Discretization published in 2001"

Strong Stability-Preserving High-Order Time Discretization Methods

Abstract: In this paper we review and further develop a class of strong stability-preserving (SSP) high-order time discretizations for semidiscrete method of lines approximations of partial differential equations. Previously termed TVD (total variation diminishing) time discretizations, these high-order time discretization methods preserve the strong stability properties of first-order Euler time stepping and have proved very useful, especially in solving hyperbolic partial differential equations. The new developments in this paper include the construction of optimal explicit SSP linear Runge--Kutta methods, their application to the strong stability of coercive approximations, a systematic study of explicit SSP multistep methods for nonlinear problems, and the study of the SSP property of implicit Runge--Kutta and multistep methods.

Review Article Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems

Abstract: In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobilike equations.

A stabilized conforming nodal integration for Galerkin mesh-free methods

Abstract: Domain integration by Gauss quadrature in the Galerkin mesh-free methods adds considerable complexity to solution procedures. Direct nodal integration, on the other hand, leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes. A strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration. For convergence, an integration constraint (IC) is introduced as a necessary condition for a linear exactness in the mesh-free Galerkin approximation. The gradient matrix of strain smoothing is shown to satisfy IC using a divergence theorem. No numerical control parameter is involved in the proposed strain smoothing stabilization. The numerical results show that the accuracy and convergent rates in the mesh-free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method. It is also demonstrated that the Gauss integration method fails to meet IC in mesh-free discretization. For this reason the proposed method provides even better accuracy than Gauss integration for Galerkin mesh-free method as presented in several numerical examples. Copyright © 2001 John Wiley & Sons, Ltd.

Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media

Abstract: We present and analyze a perfectly matched, absorbing layer model for the velocity-stress formulation of elastodynamics. The principal idea of this method consists of introducing an absorbing layer in which we decompose each component of the unknown into two auxiliary components: a component orthogonal to the boundary and a component parallel to it. A system of equations governing these new unknowns then is constructed. A damping term finally is introduced for the component orthogonal to the boundary. This layer model has the property of generating no reflection at the interface between the free medium and the artificial absorbing medium. In practice, both the boundary condition introduced at the outer boundary of the layer and the dispersion resulting from the numerical scheme produce a small reflection which can be controlled even with very thin layers. As we will show with several experiments, this model gives very satisfactory results; namely, the reflection coefficient, even in the case of heterogeneous, anisotropic media, is about 1% for a layer thickness of five space discretization steps.

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.

Discrete electromagnetism with the finite integration technique

Abstract: The Finite Integration Technique (FIT) is a consistent discretization scheme for Maxwell's equations in their integral form. The resulting matrix equations of the discretized fields can be used for efficient numerical simulations on modern computers. In addition, the basic algebraic properties of this discrete electromagnetic field theory allow to analytically and algebraically prove conservation properties with respect to energy and charge of the discrete formulation and gives an explanation of the stability properties of numerical formulations in the time domain.

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.

Discretization Methods and Iterative Solvers Based on Domain Decomposition

Abstract: Discretization Techniques Based on Domain Decomposition.- 1.1 Introduction to Mortar Finite Element Methods.- 1.2 Mortar Methods with Alternative Lagrange Multiplier Spaces.- 1.2.1 An Approximation Property.- 1.2.2 The Consistency Error.- 1.2.3 Discrete Inf-sup Conditions.- 1.2.4 Examples of Lagrange Multiplier Spaces.- 1.2.4.1 The First Order Case in 2D.- 1.2.4.2 The First Order Case in 3D.- 1.2.4.3 The Second Order Case in 2D.- 1.3 Discretization Techniques Based on the Product Space.- 1.3.1 A Dirichlet-Neumann Formulation.- 1.3.2 Variational Formulations.- 1.3.3 Algebraic Formulations.- 1.4 Examples for Special Mortar Finite Element Discretizations.- 1.4.1 The Coupling of Primal and Dual Finite Elements.- 1.4.2 An Equivalent Nonconforming Formulation.- 1.4.3 Crouzeix-Raviart Finite Elements.- 1.5 Numerical Results.- 1.5.1 Influence of the Lagrange Multiplier Spaces.- 1.5.2 A Non-optimal Mortar Method.- 1.5.3 Influence of the Choice of the Mortar Side.- 1.5.4 Influence of the Jump of the Coefficients.- Iterative Solvers Based on Domain Decomposition.- 2.1 Abstract Schwarz Theory.- 2.1.1 Additive Schwarz Methods.- 2.1.2Multiplicative Schwarz Methods.- 2.1.3 Multigrid Methods.- 2.2 Vector Field Discretizations.- 2.2.1 Raviart-Thomas Finite Elements.- 2.2.2 An Iterative Substructuring Method.- 2.2.2.1 An Interpolation Operator onto VH.- 2.2.2.2 An Extension Operator onto VF.- 2.2.2.3 Quasi-optimal Bounds.- 2.2.3 A Hierarchical Basis Method.- 2.2.3.1 Horizontal Decomposition.- 2.2.3.2 Vertical Decomposition.- 2.2.4 Numerical Results.- 2.2.4.1 The 2D Case.- 2.2.4.2 The 3D Case.- 2.3 A Multigrid Method for the Mortar Product Space Formulation.- 2.3.1 Bilinear Forms.- 2.3.2 An Approximation Property.- 2.3.3 Smoothing and Stability Properties.- 2.3.4 Implementation of the Smoothing Step.- 2.3.5 Numerical Results in 2D and 3D.- 2.3.6 Extensions to Linear Elasticity.- 2.3.6.1 Uniform Ellipticity.- 2.3.6.2 Numerical Results.- 2.3.6.3 A Weaker Interface Condition.- 2.4 A Dirichlet-Neumann Type Method.- 2.4.1 The Algorithm.- 2.4.2 Numerical Results.- 2.5 A Multigrid Method for the Mortar Saddle Point Formulation.- 2.5.1 An Approximation Property.- 2.5.2 Smoothing and Stability Properties.- 2.5.2.1 A Block Diagonal Smoother.- 2.5.2.2 An Indefinite Smoother.- 2.5.3 Numerical Results.- List of Figures.- List of Tables.- Notations.

Modeling multiphase non-isothermal fluid flow and reactive geochemical transport in variably saturated fractured rocks: 1. methodology

Abstract: Reactive fluid flow and geochemical transport in unsaturated fractured rocks have received increasing attention for studies of contaminant transport, ground- water quality, waste disposal, acid mine drainage remediation, mineral deposits, sedimentary diagenesis, and fluid-rock interactions in hydrothermal systems. This paper presents methods for modeling geochemical systems that emphasize: (1) involvement of the gas phase in addition to liquid and solid phases in fluid flow, mass transport, and chemical reactions; (2) treatment of physically and chemically heterogeneous and fractured rocks, (3) the effect of heat on fluid flow and reaction properties and processes, and (4) the kinetics of fluid-rock interaction. The physical and chemical process model is embodied in a system of partial differential equations for flow and transport, coupled to algebraic equations and ordinary differential equations for chemical interactions. For numerical solution, the continuum equations are discretized in space and time. Space discretization is based on a flexible integral finite difference approach that can use irregular gridding to model geologic structure; time is discretized fully implicitly as a first-order finite difference. Heterogeneous and fractured media are treated with a general multiple interacting continua method that includes double-porosity, dual-permeability, and multi-region models as special cases. A sequential iteration approach is used to treat the coupling between fluid flow and mass transport on the one hand, chemical reactions on the other. Applications of the methods developed here to variably saturated geochemical systems are presented in a companion paper (part 2, this issue).

Inequalities on Time Scales: A Survey

Abstract: The study of dynamic equations on time scales, which goes back to its founder Stefan Hilger (1988), is an area of mathematics which is currently receiving considerable attention. Although the basic aim of this is to unify the study of differential and difference equations, it also extends these classical cases to cases “in between”. In this paper we present time scales versions of the inequalities: Holder, Cauchy-Schwarz, Minkowski, Jensen, Gronwall, Bernoulli, Bihari, Opial, Wirtinger, and Lyapunov. 1. Unifying Continuous and Discrete Analysis In 1988, Stefan Hilger [13] introduced the calculus on time scales which unifies continuous and discrete analysis. A time scale is a closed subset of the real numbers. We denote a time scale by the symbol T . For functions y defined on T , we introduce a so-called delta derivative y∆ . This delta derivative is equal to y (the usual derivative) if T = R is the set of all real numbers, and it is equal to ∆y (the usual forward difference) if T = Z is the set of all integers. Then we study dynamic equations f (t; y; y∆; y∆ 2 ; : : : ; y∆ n ) = 0; which may involve higher order derivatives as indicated. Along with such dynamic equations we consider initial values and boundary conditions. We remark that these dynamic equations are differential equations when T = R and difference equations when T = Z . Other kinds of equations are covered by them as well, such as q difference equations, where T = q := fqkj k 2 Zg[ f0g for some q > 1 and difference equations with constant step size, where T = hZ := fhkj k 2 Zg for some h > 0: Particularly useful for the discretization purpose are time scales of the form T = ftkj k 2 Zg where tk 2 R; tk < tk+1 for all k 2 Z: Mathematics subject classification (2000): 34A40, 39A13.

A fully non‐linear model for three‐dimensional overturning waves over an arbitrary bottom

Abstract: An accurate three-dimensional numerical model, applicable to strongly non-linear waves, is proposed. The model solves fully non-linear potential flow equations with a free surface using a higher-order three-dimensional boundary element method (BEM) and a mixed Eulerian–Lagrangian time updating, based on second-order explicit Taylor series expansions with adaptive time steps. The model is applicable to non-linear wave transformations from deep to shallow water over complex bottom topography up to overturning and breaking. Arbitrary waves can be generated in the model, and reflective or absorbing boundary conditions specified on lateral boundaries. In the BEM, boundary geometry and field variables are represented by 16-node cubic ‘sliding’ quadrilateral elements, providing local inter-element continuity of the first and second derivatives. Accurate and efficient numerical integrations are developed for these elements. Discretized boundary conditions at intersections (corner/edges) between the free surface or the bottom and lateral boundaries are well-posed in all cases of mixed boundary conditions. Higher-order tangential derivatives, required for the time updating, are calculated in a local curvilinear co-ordinate system, using 25-node ‘sliding’ fourth-order quadrilateral elements. Very high accuracy is achieved in the model for mass and energy conservation. No smoothing of the solution is required, but regridding to a higher resolution can be specified at any time over selected areas of the free surface. Applications are presented for the propagation of numerically exact solitary waves. Model properties of accuracy and convergence with a refined spatio-temporal discretization are assessed by propagating such a wave over constant depth. The shoaling of solitary waves up to overturning is then calculated over a 1:15 plane slope, and results show good agreement with a two-dimensional solution proposed earlier. Finally, three-dimensional overturning waves are generated over a 1:15 sloping bottom having a ridge in the middle, thus focusing wave energy. The node regridding method is used to refine the discretization around the overturning wave. Convergence of the solution with grid size is also verified for this case. Copyright © 2001 John Wiley & Sons, Ltd.

A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation

Abstract: A compressible multiphase unconditionally hyperbolic model is proposed. It is able to deal with a wide range of applications: interfaces between compressible materials, shock waves in condensed multiphase mixtures, homogeneous two-phase flows (bubbly and droplet flows) and cavitation in liquids. Here we focus on the generalization of the formulation to an arbitrary number of fluids, and to mass and energy transfers, and extend the associated Godunov method.We first detail the specific problems involved in the computation of thermodynamic interface variables when dealing with compressible materials separated by well-defined interfaces. We then address one of the major problems in the modelling of detonation waves in condensed energetic materials and propose a way to suppress the mixture equation of state. We then consider another problem of practical importance related to high-pressure liquid injection and associated cavitating flow. This problem involves the dynamic creation of interfaces. We show that the multiphase model is able to solve these very different problems using a unique formulation.We then develop the Godunov method for this model. We show how the non-conservative terms must be discretized in order to fulfil the interface conditions. Numerical resolution of interface conditions and partial equilibrium multiphase mixtures also requires the introduction of infinite relaxation terms. We propose a way to solve them in the context of an arbitrary number of fluids. This is of particular importance for the development of multimaterial reactive hydrocodes. We finally extend the discretization method in the multidimensional case, and show some results and validations of the model and method.

A model for the quasi-static growth of a brittle fracture: existence and approximation results

Abstract: We give a precise mathematical formulation of a variational model for the irreversible quasi-static evolution of a brittle fracture proposed by G.A. Francfort and J.-J. Marigo, and based on Griffith's theory of crack growth. In the two-dimensional case we prove an existence result for the quasi-static evolution and show that the total energy is an absolutely continuous function of time, although we can not exclude that the bulk energy and the surface energy may present some jump discontinuities. This existence result is proved by a time discretization process, where at each step a global energy minimization is performed, with the constraint that the new crack contains all cracks formed at the previous time steps. This procedure provides an effective way to approximate the continuous time evolution.

A two-grid discretization scheme for eigenvalue problems

Abstract: A two-grid discretization scheme is proposed for solving eigenvalue problems, including both partial differential equations and integral equations. With this new scheme, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid, and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy.

A Multigrid Method Enhanced by Krylov Subspace Iteration for Discrete Helmholtz Equations

Abstract: Standard multigrid algorithms have proven ineffective for the solution of discretizations of Helmholtz equations. In this work we modify the standard algorithm by adding GMRES iterations at coarse levels and as an outer iteration. We demonstrate the algorithm's effectiveness through theoretical analysis of a model problem and experimental results. In particular, we show that the combined use of GMRES as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as 200. For fixed wave numbers, it displays grid-independent convergence rates and has costs proportional to the number of unknowns.

A further refinement of discretized Lyapunov functional method for the stability of time-delay systems

Abstract: The discretized Lyapunov functional method for the stability problem of time-delay systems is further refined using a combination of integral inequality and variable elimination technique. As a result, the computational requirement is further reduced for the same discretization mesh. For systems without uncertainty, the convergence to the analytical solution is greatly accelerated. For uncertain systems, the new method is much less conservative. Numerical examples are presented to illustrate the effectiveness of the method.

Finite element discretization of the Navier–Stokes equations with a free capillary surface

Abstract: The instationary Navier–Stokes equations with a free capillary boundary are considered in 2 and 3 space dimensions. A stable finite element discretization is presented. The key idea is the treatment of the curvature terms by a variational formulation. In the context of a discontinuous in time space–time element discretization stability in (weak) energy norms can be proved. Numerical examples in 2 and 3 space dimensions are given.

Structural Acoustics and Vibration

Abstract: This book is devoted to mechanical models, variational formulations and discretization for calculating linear vibrations in the frequency domain of complex structures with arbitrary shape, coupled or not with external and internal acoustic fluids at rest. Such coupled systems are encountered in the area of internal and external noise prediction, reduction and control problems. The excitations can arise from different mechanisms such as mechanical forces applied to the structure, internal acoustic sources, external acoustic sources and external incident acoustic plane waves. These excitations can be deterministic or random. We are interested not only in the low-frequency domain for which modal analysis is suitable, but also in the medium-frequency domain for which additional mechanical modeling and appropriate solving methods are necessary. The main objective of the book is to present appropriate theoretical formulations, constructed so as to be directly applicable for developing computer codes for the numerical simulation of complex systems.

Discrete Electromagnetism With the Finite Integration Technique - Abstract

Abstract: The Finite Integration Technique (FIT) is a consistent discretization scheme for Maxwell's equations in their integral form. The resulting matrix equations of the discretized fields can be used for efficient numerical simulations on modern computers. In addition, the basic algebraic properties of this discrete electromagnetic field theory allow to analytically and algebraically prove conservation properties with respect to energy and charge of the discrete formulation and gives an explanation of the stability properties of numerical time domain formulations.

Berry-phase treatment of the homogeneous electric field perturbation in insulators

Abstract: A perturbation theory of the static response of insulating crystals to homogeneous electric fields that combines the modem theory of polarization (MTP) with the variation-perturbation framework is developed at unrestricted order of perturbation. First, we address conceptual issues related to the definition of such a perturbative approach. In particular, in our definition of an electric-field-dependent energy functional for periodic systems, the position operator appearing in the perturbation term is replaced by a Berry-phase expression, along the lines of the MTP. Moreover, due to the unbound nature of the perturbation, a regularization of the Ferry-phase expression for the polarization is needed in order to define a numerically stable variational procedure. Regularization is achieved by means of discretization, which can be performed either before or after the perturbation expansion. We compare the two possibilities and apply them to a model tight-binding Hamiltonian. Lowest-order as well as generic formulas are presented for the derivatives of the total energy, the normalization condition, the eigenequation, and the Lagrange parameters.

On the modelling of the subgrid-scale and filtered-scale stress tensors in large-eddy simulation

Abstract: The large-eddy simulation (LES) equations are obtained from the application of two operators to the Navier{Stokes equations: a smooth lter and a discretization operator. The introduction ab initio of the discretization influences the structure of the unknown stress in the LES equations, which now contain a subgrid-scale stress tensor mainly due to discretization, and a filtered-scale stress tensor mainly due to filtering. Theoretical arguments are proposed supporting eddy viscosity models for the subgrid-scale stress tensor. However, no exact result can be derived for this term because the discretization is responsible for a loss of information and because its exact nature is usually unknown. The situation is different for the filtered-scale stress tensor for which an exact expansion in terms of the large-scale velocity and its derivatives is derived for a wide class of filters including the Gaussian, the tophat and all discrete filters. As a consequence of this generalized result, the filtered-scale stress tensor is shown to be invariant under the change of sign of the large-scale velocity. This implies that the filtered-scale stress tensor should lead to reversible dynamics in the limit of zero molecular viscosity when the discretization effects are neglected. Numerical results that illustrate this effect are presented together with a discussion on other approaches leading to reversible dynamics like the scale similarity based models and, surprisingly, the dynamic procedure.