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Showing papers on "Numerical analysis published in 1994"


Book
01 Aug 1994
TL;DR: In this article, the authors provide a thorough illustration of numerical methods, carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation.
Abstract: This is the softcover reprint of the very popular hardcover edition. This book deals with the numerical approximation of partial differential equations. Its scope is to provide a thorough illustration of numerical methods, carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation. A sound balancing of theoretical analysis, description of algorithms and discussion of applications is one of its main features. Many kinds of problems are addressed. A comprehensive theory of Galerkin method and its variants, as well as that of collocation methods, are developed for the spatial discretization. These theories are then specified to two numerical subspace realizations of remarkable interest: the finite element method and the spectral method.

2,383 citations


Journal ArticleDOI
TL;DR: This paper establishes that the interior-reflective Newton approach is globally and quadratically convergent, and develops a specific example of interior- reflective Newton methods which can be used for large-scale and sparse problems.
Abstract: We consider a new algorithm, an interior-reflective Newton approach, for the problem of minimizing a smooth nonlinear function of many variables, subject to upper and/or lower bounds on some of the variables. This approach generatesstrictly feasible iterates by using a new affine scaling transformation and following piecewise linear paths (reflection paths). The interior-reflective approach does not require identification of an "activity set". In this paper we establish that the interior-reflective Newton approach is globally and quadratically convergent. Moreover, we develop a specific example of interior-reflective Newton methods which can be used for large-scale and sparse problems.

1,101 citations


Journal ArticleDOI
TL;DR: In this paper, a method for constructing boundary conditions (numerical and physical) of the required accuracy for compact (Pade-like) high-order finite-difference schemes for hyperbolic systems is presented.

728 citations


Book
01 Jan 1994
TL;DR: In this paper, the effects of space discretization on wave propagation are investigated and a detailed treatment of boundary conditions is given. But the results are limited to three-dimensional shallow-water flows.
Abstract: Preface. 1. Shallow-water flows. 2. Equations. 3. Some properties. 4. Behaviour of solutions. 5. Boundary conditions. 6. Discretization in space. 7. Effect of space discretization on wave propagation. 8. Time integration methods. 9. Effects of time discretization on wave propagation. 10. Numerical treatment of boundary conditions. 11. Three-dimensional shallow-water flow. List of notations. References. Index.

527 citations


01 Apr 1994
TL;DR: In this article, the elastic problem for a heterogeneous material is formulated with the help of a homogeneous reference medium and written under the form of a periodic Lippman-Schwinger equation.
Abstract: This Note is devoted to a new iterative algorithm to compute the local and overall response of a composite from images of its (complex) microstructure. The elastic problem for a heterogeneous material is formulated with the help of a homogeneous reference medium and written under the form of a periodic Lippman-Schwinger equation. Using the fact that the Green's function of the pertinent operator is known explicitely in Fourier space, this equation is solved iteratively.The method is extended to the case where the individual constituents are elastic-plastic Von Mises materials with isotropic hardening

427 citations


Journal ArticleDOI
TL;DR: In this paper, the Navier-Stokes equations for constant viscosity were solved using the SPH method and the expected parabolic and paraboloid velocity profiles were obtained.
Abstract: present a new SPH method that can be used to solve the Navier-Stokes equations for constant viscosity. The method is applied to two-dimensional Poiseuille flow, three-dimensional Hagen­ Poiseuille flow and two-dimensional isothermal flows around a cylinder. In the former two cases, the temperature of fluid is assumed to be linearly dependent on a coordinate variable x along the flow direction. The numerical results agree well with analytic solutions, and we obtain nearly uniform density distributions and the expected parabolic and paraboloid velocity profiles. The density and ·velocity field in the latter case are compared with the results obtained using a finite difference method. Both methods give similar results for Reynolds number Re=6, 10, 20, 30 and 55, and the differences in the total drag coefficients are about 2~4%. Our numerical simulations indicate that SPH is also an effective numerical method for calculation of viscous flows.

369 citations


Journal ArticleDOI
TL;DR: A multilevel algorithm is applied to the solution of an integral equation using the conjugate gradient method and shows that the complexity of a matrix-vector multiplication is proportional to N (log(N))2.
Abstract: In the solution of an integral equation using the conjugate gradient (CG) method, the most expensive part is the matrix-vector multiplication, requiring O(N2) floating-point operations. The fast multipole method (FMM) reduced the operation to O(N15). In this article we apply a multilevel algorithm to this problem and show that the complexity of a matrix-vector multiplication is proportional to N (log(N))2. © 1994 John Wiley & Sons, Inc.

303 citations


Journal ArticleDOI
01 Nov 1994
TL;DR: In this paper, a high-precision numerical time step integration method is proposed for a linear time-invariant structural dynamic system, which is almost identical to the precise solution and is independent of the time step size for a wide range of step sizes.
Abstract: A high-precision numerical time step integration method is proposed for a linear time-invariant structural dynamic system. Its numerical results are almost identical to the precise solution and are almost independent of the time step size for a wide range of step sizes. Numerical examples illustrate this high precision.

285 citations


Journal ArticleDOI
Yutaka Sasaki1
TL;DR: Comparisons of numerical examples show that the full inversion method gives a better resolution, particularly for the near-surface features, than does the approximate method, since the full derivatives are more sensitive to local features of resistivity variations than are the approximate derivatives.
Abstract: With the increased availability of faster computers, it is now practical to employ numerical modeling techniques to invert resistivity data for 3-D structure. Full and approximate 3-D inversion methods using the finite-element solution for the forward problem have been developed. Both methods use reciprocity for efficient evaluations of the partial derivatives of apparent resistivity with respect to model resistivities. In the approximate method, the partial derivatives are approximated by those for a homogeneous half-space, and thus the computation time and memory requirement are further reduced. The methods are applied to synthetic data sets from 3-D models to illustrate their effectiveness. They give a good approximation of the actual 3-D structure after several iterations in practical situations where the effects of model inadequacy and topography exist. Comparisons of numerical examples show that the full inversion method gives a better resolution, particularly for the near-surface features, than does the approximate method. Since the full derivatives are more sensitive to local features of resistivity variations than are the approximate derivatives, the resolution of the full method may be further improved when the finite-element solutions are performed more accurately and more efficiently.

277 citations


Journal ArticleDOI
TL;DR: In this paper, the authors developed plane strain boundary equations for axisymmetric shear and dilation waves based on an approximation of the form of the outward traveling waves, which are equivalent to mechanical systems with frequency independent components.
Abstract: Finite element analysis of dynamic foundation problems requires the use of transmitting boundaries to model the radiation of waves from the finite element mesh into the far field. Problems involving inelastic behavior of the soil in the near field are most readily solved in the time domain. The standard viscous boundary is widely used in such situations. However, in axisymmetric situations this boundary is inappropriate. This paper develops plane strain boundary equations for axisymmetric shear and dilation waves based on an approximation of the form of the outward traveling waves. These boundary equations are shown to be equivalent to mechanical systems with frequency independent components. The complex stiffnesses of the new boundaries are compared with the equivalent viscous and plane strain boundary stiffnesses, and the new boundaries are found to agree closely with the plane strain boundaries. The response of an extended axisymmetric finite element mesh subjected to a transient force of the type gene...

265 citations


Journal ArticleDOI
TL;DR: In this paper, a spectral element method for studying acoustic wave propagation in complex geological structures is presented, which shows more accurate results compared to the low-order finite element, the conventional finite difference and the pseudospectral methods.

Journal ArticleDOI
TL;DR: In this paper, convergence and stability bounds for a class of time-discrete and fully discrete approximation schemes for boundary integral equations of such problems were derived for the single-layer potential equation of the wave equation.
Abstract: Convergence estimates in terms of the data are shown for multistep methods applied to non-homogeneous linear initial-boundary value problems. Similar error bounds are derived for a new class of time-discrete and fully discrete approximation schemes for boundary integral equations of such problems, e.g., for the single-layer potential equation of the wave equation. In both cases, the results are obtained from convergence and stability estimates for operational quadrature approximations of convolutions. These estimates, which are also proved here, depend on bounds of the Laplace transform of the (distributional) convolution kernel outside the stability region scaled by the time stepsize, and on the smoothness of the data.

Journal ArticleDOI
TL;DR: In this article, a path-following technique is presented for the numerical solution of a class of elastic structural problems, which is based on applying a perturbation technique in a stepwise manner.

Journal ArticleDOI
TL;DR: In this paper, the authors give a proof of Adomian's method using only some properties of the nonlinear function, and apply these results to some concrete problems, such as the problem of finding a nonlinear solution to a set of nonlinear problems.

Journal Article
TL;DR: An approximate Riemann solver for the governing equations of ideal magnetohydrodynamics (MHD) was developed in this article, which has an eight-wave structure, where seven of the waves are those used in previous work on upwind schemes for MHD, and the eighth wave is related to the divergence of the magnetic field.
Abstract: An approximate Riemann solver is developed for the governing equations of ideal magnetohydrodynamics (MHD). The Riemann solver has an eight-wave structure, where seven of the waves are those used in previous work on upwind schemes for MHD, and the eighth wave is related to the divergence of the magnetic field. The structure of the eighth wave is not immediately obvious from the governing equations as they are usually written, but arises from a modification of the equations that is presented in this paper. The addition of the eighth wave allows multi-dimensional MHD problems to be solved without the use of staggered grids or a projection scheme, one or the other of which was necessary in previous work on upwind schemes for MHD. A test problem made up of a shock tube with rotated initial conditions is solved to show that the two-dimensional code yields answers consistent with the one-dimensional methods developed previously.


Journal ArticleDOI
TL;DR: In this article, the non-linear branches are sought in the form of asymptotic expansions, and they are determined by solving numerically (FEM) several linear problems with a single stiffness matrix.
Abstract: In this paper, we apply asymptotic–numerical methods for computing non-linear equilibrium paths of elastic beam, plate and shell structures. The non-linear branches are sought in the form of asymptotic expansions, and they are determined by solving numerically (FEM) several linear problems with a single stiffness matrix. A large number of terms of the series can be easily computed by using recurrence formulas. In comparison with a more classical step-by-step procedure, the method is rapid and automatic. We show, with some examples, that the choice of the expansion's parameter and the use of Pade approximants play an important role in the determination of the size of the domain of convergence.

Journal ArticleDOI
TL;DR: Several versions of a moving mesh method are developed based on a mesh spatial smoothing technique and on the moving mesh PDEs derived in a previous paper, which clearly demonstrate that the present methods are capable of accurately tracking rapid spatial and temporal transitions.

Journal ArticleDOI
TL;DR: In this paper, a computational procedure is developed for predicting separated turbulent flows in complex two-dimensional and three-dimensional geometries, based on the fully conservative, structured finite volume framework within which the volumes are non-orthogonal and collocated such that all flow variables are stored at one and the same set of nodes.

Journal ArticleDOI
TL;DR: In this paper, a numerical method for accurate simulation of the inner and outer vortex structures in transitional H 2 /N 2 jet diffusion flames is presented, incorporating buoyancy, a simple one-step chemistry model, coefficients that depend on temperature and species concentration.
Abstract: A numerical method for accurate simulation of the time and spatial characteristics of the inner and outer vortex structures in transitional H 2 /N 2 jet diffusion flames is presented. The direct numerical simulation, incorporating buoyancy, a simple one-step chemistry model, coefficients that depend on temperature and species concentration, is described in detail. The species and energy equations are simplified by introducing two conserved scalars β 1 and β 2 and by assuming that the Lewis number of the flow is equal to unity. An implicit, third-order-accurate, upwind numerical scheme having very low numerical diffusion is used to simulate the inner small-scale structures and the outer large-scale structures simultaneously

Journal ArticleDOI
TL;DR: In this article, the exact renormalization group or flow equation for the effective action and its decomposition into one particle irreducibleN point functions is discussed and a combination of analytic and numerical methods is proposed, which is applied to the Wick-Cutkosky model and a QCD-motivated interaction.
Abstract: We discuss the exact renormalization group or flow equation for the effective action and its decomposition into one particle irreducibleN point functions. With the help of a truncated flow equation for the four point function we study the bound state problem for scalar fields. A combination of analytic and numerical methods is proposed, which is applied to the Wick-Cutkosky model and a QCD-motivated interaction. We present results for the bound state masses and the Bethe-Salpeter wave function.

BookDOI
01 Jan 1994
TL;DR: Invited Talks.- Non-matching Grids and Lagrange Multipliers.- A FETI Method for a Class of Indefinite or Complex Second- or Fourth-Order Problems.
Abstract: Invited Talks.- Non-matching Grids and Lagrange Multipliers.- A FETI Method for a Class of Indefinite or Complex Second- or Fourth-Order Problems.- Hybrid Schwarz-Multigrid Methods for the Spectral Element Method: Extensions to Navier-Stokes.- Numerical Approximation of Dirichlet-to-Neumann Mapping and its Application to Voice Generation Problem.- Selecting Constraints in Dual-Primal FETI Methods for Elasticity in Three Dimensions.- Coupled Boundary and Finite Element Tearing and Interconnecting Methods.- Parallel Simulation of Multiphase/Multicomponent Flow Models.- Uncoupling-Coupling Techniques for Metastable Dynamical Systems.- Minisymposium: Domain Decomposition Methods for Wave Propagation in Unbounded Media.- On the Construction of Approximate Boundary Conditions for Solving the Interior Problem of the Acoustic Scattering Transmission Problem.- Approximation and Fast Calculation of Non-local Boundary Conditions for the Time-dependent Schrodinger Equation.- Domain Decomposition and Additive Schwarz Techniques in the Solution of a TE Model of the Scattering by an Electrically Deep Cavity.- Minisymposium: Parallel Finite Element Software.- A Model for Parallel Adaptive Finite Element Software.- Towards a Unified Framework for Scientific Computing.- Distributed Point Objects. A New Concept for Parallel Finite Elements.- Minisymposium: Collaborating Subdomains for Multi-Scale Multi-Physics Modelling.- Local Defect Correction Techniques Applied to a Combustion Problem.- Electronic Packaging and Reduction in Modelling Time Using Domain Decomposition.- Improving Robustness and Parallel Scalability of Newton Method Through Nonlinear Preconditioning.- Iterative Substructuring Methods for Indoor Air Flow Simulation.- Fluid-Structure Interaction Using Nonconforming Finite Element Methods.- Interaction Laws in Viscous-Inviscid Coupling.- Minisymposium: Recent Developments for Schwarz Methods.- Comparison of the Dirichlet-Neumann and Optimal Schwarz Method on the Sphere.- Finite Volume Methods on Non-Matching Grids with Arbitrary Interface Conditions and Highly Heterogeneous Media.- Nonlinear Advection Problems and Overlapping Schwarz Waveform Relaxation.- A New Cement to Glue Nonconforming Grids with Robin Interface Conditions: The Finite Element Case.- Acceleration of a Domain Decomposition Method for Advection-Diffusion Problems.- A Stabilized Three-Field Formulation and its Decoupling for Advection-Diffusion Problems.- Approximation of Optimal Interface Boundary Conditions for Two-Lagrange Multiplier FETI Method.- Optimized Overlapping Schwarz Methods for Parabolic PDEs with Time-Delay.- Minisymposium: Trefftz-Methods.- A More General Version of the Hybrid-Trefftz Finite Element Model by Application of TH-Domain Decomposition.- Minisymposium: Domain Decomposition on Nonmatching Grids.- Mixed Finite Element Methods for Diffusion Equations on Nonmatching Grids.- Mortar Finite Elements with Dual Lagrange Multipliers: Some Applications.- Non-Conforming Finite Element Methods for Nonmatching Grids in Three Dimensions.- On an Additive Schwarz Preconditioner for the Crouzeix-Raviart Mortar Finite Element.- Minisymposium: FETI and Neumann-Neumann Domain Decomposition Methods.- A FETI-DP Method for the Mortar Discretization of Elliptic Problems with Discontinuous Coefficients.- A FETI-DP Formulation for Two-dimensional Stokes Problem on Nonmatching Grids.- Some Computational Results for Dual-Primal FETI Methods for Elliptic Problems in 3D.- The FETI Based Domain Decomposition Method for Solving 3D-Multibody Contact Problems with Coulomb Friction.- Choosing Nonmortars: Does it Influence the Performance of FETI-DP Algorithms?.- Minisymposium: Heterogeneous Domain Decomposition with Applications in Multiphysics.- Domain Decomposition Methods in Electrothermomechanical Coupling Problems.- A Multiphysics Strategy for Free Surface Flows.- Minisymposium: Robust Decomposition Methods for Parameter Dependent Problems.- Weighted Norm-Equivalences for Preconditioning.- Preconditioning for Heterogeneous Problems.- Minisymposium: Recent Advances for the Parareal in Time Algorithm.- On the Convergence and the Stability of the Parareal Algorithm to Solve Partial Differential Equations.- A Parareal in Time Semi-implicit Approximation of the Navier-Stokes Equations.- The Parareal in Time Iterative Solver: a Further Direction to Parallel Implementation.- Stability of the Parareal Algorithm.- Minisymposium: Space Decomposition and Subspace Correction Methods for Linear and Nonlinear Problems.- Multilevel Homotopic Adaptive Finite Element Methods for Convection Dominated Problems.- A Convergent Algorithm for Time Parallelization Applied to Reservoir Simulation.- Nonlinear Positive Interpolation Operators for Analysis with Multilevel Grids.- Minisymposium: Discretization Techniques and Algorithms for Multibody Contact Problems.- On Scalable Algorithms for Numerical Solution of Variational Inequalities Based on FETI and Semi-monotonic Augmented Lagrangians.- Fast Solving of Contact Problems on Complicated Geometries.- Contributed Talks.- Generalized Aitken-like Acceleration of the Schwarz Method.- The Fat Boundary Method: Semi-Discrete Scheme and Some Numerical Experiments.- Modelling of an Underground Waste Disposal Site by Upscaling and Simulation with Domain Decomposition Method.- Non-Overlapping DDMs to Solve Flow in Heterogeneous Porous Media.- Domain Embedding/Controllability Methods for the Conjugate Gradient Solution of Wave Propagation Problems.- An Accelerated Block-Parallel Newton Method via Overlapped Partitioning.- Generation of Balanced Subdomain Clusters with Minimum Interface for Distributed Domain Decomposition Applications.- Iterative Methods for Stokes/Darcy Coupling.- Preconditioning Techniques for the Bidomain Equations.- Direct Schur Complement Method by Hierarchical Matrix Techniques.- Balancing Neumann-Neumann Methods for Elliptic Optimal Control Problems.- Domain Decomposition Preconditioners for Spectral Nedelec Elements in Two and Three Dimensions.- Parallel Distributed Object-Oriented Framework for Domain Decomposition.- A Domain Decomposition Based Two-Level Newton Scheme for Nonlinear Problems.- Domain Decomposition for Discontinuous Galerkin Method with Application to Stokes Flow.- Hierarchical Matrices for Convection-Dominated Problems.- Parallel Performance of Some Two-Level ASPIN Algorithms.- Algebraic Analysis of Schwarz Methods for Singular Systems.- Schwarz Waveform Relaxation Method for the Viscous Shallow Water Equations.- A Two-Grid Alternate Strip-Based Domain Decomposition Strategy in Two-Dimensions.- Parallel Solution of Cardiac Reaction-Diffusion Models.- Predictor-Corrector Methods for Solving Continuous Casting Problem.

Journal ArticleDOI
TL;DR: In this paper, a simple and practical numerical method for the liquefaction analysis is formulated using au-p (displacement of the solid phase-pore water pressure) formulation, and the accuracy of the proposed numerical method is addressed through a comparison of the numerical results and the analytical solutions for the transient response of saturated porous solids.
Abstract: The phenomenon of liquefaction is one of the most important subjects in Earthquake Engineering and Coastal Engineering. In the present study, the governing equations of such coupling problems as soil skeleton and pore water are obtained through application of the two-phase mixture theory. Using au-p (displacement of the solid phase-pore water pressure) formulation, a simple and practical numerical method for the liquefaction analysis is formulated. The finite difference method (FDM) is used for the spatial discretization of the continuity equation to define the pore water pressure at the center of the element, while the finite element method (FEM) is used for the spatial discretization of the equilibrium equation. FEM-FDM coupled analysis succeeds in reducing the degrees of freedom in the descretized equations. The accuracy of the proposed numerical method is addressed through a comparison of the numerical results and the analytical solutions for the transient response of saturated porous solids. An elasto-plastic constitutive model based on the non-linear kinematic hardening rule is formulated to describe the stress-strain behavior of granular materials under cyclic loading. Finally, the applicability of the proposed numerical method is examined. The following two numerical examples are analyzed in this study: (1) the behavior of seabed deposits under wave action, and (2) a numerical simulation of shaking table test of coal fly ash deposit.

Journal ArticleDOI
TL;DR: In this article, a general method for constructing high-order approximation schemes for Hamilton-Jacobi-Bellman equations is given, based on a discrete version of the Dynamic Programming Principle.
Abstract: A general method for constructing high-order approximation schemes for Hamilton-Jacobi-Bellman equations is given. The method is based on a discrete version of the Dynamic Programming Principle. We prove a general convergence result for this class of approximation schemes also obtaining, under more restrictive assumptions, an estimate in $L^\infty$ of the order of convergence and of the local truncation error. The schemes can be applied, in particular, to the stationary linear first order equation in ${\Bbb R}^n$ . We present several examples of schemes belonging to this class and with fast convergence to the solution.

Journal ArticleDOI
TL;DR: This paper presents the first endeavour to exploit a generalized differential quadrature method as an accurate, efficient and simple numerical technique for structural analysis as well as seeking an alternate numerical method using fewer grid points to find results with acceptable accuracy.
Abstract: SUMMARY This paper presents the first endeavour to exploit a generalized differential quadrature method as an accurate, efficient and simple numerical technique for structural analysis. Firstly, drawbacks existing in the method of differential quadrature (DQ) are evaluated and discussed. Then, an improved and simpler generalized differential quadrature method (GDQ) is introduced to overcome the existing drawback and to simplify the procedure for determining the weighting coefficients. Subsequently, the generalized differential quadrature is systematically employed to solve problems in structural analysis. Numerical examples have shown the superb accuracy, efficiency, convenience and the great potential of this method. Numerical approximation methods for solving partial differential equations have been widely used in various engineering fields. Classical techniques such as finite element and finite difference methods are well developed and well known. These methods can provide very accurate results by using a large number of grid points. However, in a large number of cases, reasonably approximate solutions are desired at only a few specific points in the physical domain. In order to get results even at or around a point of interest with acceptable accuracy, conventional finite element and finite difference methods still require the use of a large number of grid points. Consequently, the requirement for computer capacity is often unnecessarily large in such cases. In seeking an alternate numerical method using fewer grid points to find results with acceptable accuracy, the method of differential quadrature (DQ) was introduced by Bellman et a!.'. The method of DQ is a global approximate method. This method is based on the ideas that the derivative of a function with respect to a co-ordinate direction can be expressed as a weighted linear sum of all the function values at all mesh points along that direction and that a continuous function can be approximated by a higher-order polynomial in the overall domain. The DQ method differs from the finite element method (FEM) in two aspects. Firstly, the FEM uses lower-order polynomials to approximate a function on a local element level, while the DQ method approximates a function on the global area using higher-order polynomials. Secondly, the DQ method directly approximates the derivatives of a function at a point, while the FEM approximates a function over a local element and the derivatives can then be derived from the approximate function. In this aspect, the DQ method is more similar to the finite difference method (FDM). However, the FDM is also a local approximation method based on lower-order polynomial approximation. In fact, it can be shown that the FDM is just a special case of the DQ method where it is applied locally on the range [.xi- xi + l]. Owing to the higher-order

Journal ArticleDOI
TL;DR: A new iterative scheme for the numerical solution of a class of linear variational inequalities and the convergence proof are introduced.
Abstract: In this paper we introduce a new iterative scheme for the numerical solution of a class of linear variational inequalities. Each iteration of the method consists essentially only of a projection to a closed convex set and two matrix-vector multiplications. Both the method and the convergence proof are very simple.

Book ChapterDOI
01 Jan 1994
TL;DR: The singular value decomposition of matrices is widely used in least squares estimation, systems approximations, and numerical linear algebra.
Abstract: Many numerical methods used in application areas such as signal processing, estimation, and control are based on the singular value decomposition (SVD) of matrices. The SVD is widely used in least squares estimation, systems approximations, and numerical linear algebra.

Journal ArticleDOI
TL;DR: In this paper, the Yee scheme is shown to be second-order convergent on a non-uniform mesh, despite the fact that the local truncation error is only of first order.
Abstract: The Yee scheme is the principal finite difference method used in computing time domain solutions of Maxwell's equations. On a uniform grid the method is easily seen to be second- order convergent in space. This paper shows that the Yee scheme is also second-order convergent on a nonuniform mesh despite the fact that the local truncation error is (nodally) only of first order.

Journal ArticleDOI
TL;DR: A subclass of algorithms which retain these strong notions of nonlinear stability and long-term dissipative behavior is identified which, in addition, has the remarkable property of being linear within the time step.

Journal ArticleDOI
TL;DR: In this article, the second-order terms associated with geometric nonlinearity were introduced into the basic equation of generalized beam theory to give rise to simple explicit equations for the load to cause buckling in individual modes under either axial load or uniform bending moment.