Abstract: Explicit finite element and finite difference methods are used to solve a wide variety of transient problems in industry and academia. Unfortunately, explicit methods are rarely discussed in detail in finite element text books. This paper reviews the basic explicit finite element and finite difference methods that are currently used to solve transient, large deformation problems in solid mechanics. A special emphasis has been placed on documenting methods that have not been previously published in journals.

Abstract: The finite element method (FEM) has grown from a civil engineering tool into a general method for solving partial differential equations. In this area it beats its competitors: the finite difference method (FDM) and the finite volume method (FVM), in that it is better suited to deal with complex geometries and difficult boundary conditions. As opposed to that, it usually is more difficult to apply and the resulting sets of equations have a more complicated structure.

Abstract: A semi-implicit finite difference method for the numerical solution of three-dimensional shallow water flows is presented and discussed. The governing equations are the primitive three-dimensional turbulent mean flow equations where the pressure distribution in the vertical has been assumed to be hydrostatic. In the method of solution a minimal degree of implicitness has been adopted in such a fashion that the resulting algorithm is stable and gives a maximal computational efficiency at a minimal computational cost. At each time step the numerical method requires the solution of one large linear system which can be formally decomposed into a set of small three-diagonal systems coupled with one five-diagonal system. All these linear systems are symmetric and positive definite. Thus the existence and uniquencess of the numerical solution are assured. When only one vertical layer is specified, this method reduces as a special case to a semi-implicit scheme for solving the corresponding two-dimensional shallow water equations. The resulting two- and three-dimensional algorithm has been shown to be fast, accurate and mass-conservative and can also be applied to simulate flooding and drying of tidal mud-flats in conjunction with three-dimensional flows. Furthermore, the resulting algorithm is fully vectorizable for an efficient implementation on modern vector computers.

Abstract: The finite-difference-time-domain (FDTD) method is generalized to include the accurate modeling of curved surfaces. This generalization, the contour path CP), method, accurately models the illumination of bodies with curved surfaces, yet retains the ability to model corners and edges. CP modeling of two-dimensional electromagnetic wave scattering from objects of various shapes and compositions is presented. >

Abstract: The newly developed finite-difference vector beam propagation method (FD-VBPM) is analyzed and assessed for application to two-dimensional waveguide structures. The general formulations for the FD-VBPM are derived from the vector wave equations for the electric fields. The stability criteria, the numerical dissipation, and the dispersion of the finite-difference schemes are analyzed by applying the von Neumann method. Important issues regarding the implementation, such as the choice of reference refractive index, the application of numerical boundary conditions, and the use of numerical solution schemes, are discussed. The FD-VBPM is assessed by calculating the attenuation coefficients and the percentage errors of the propagation constants of the TE and TM modes of a step-index slab waveguide. Several salient features of the FD-VBPM are illustrated. >

Abstract: The main objective of this survey is to study convergence properties of difference methods applied to differential inclusions. It presents, in a unified way, a number of results scattered in the li...

Abstract: We demonstrate that relaxation algorithms for the determination of the lowest-order modes of a refractive-index profile are identical to imaginary-distance propagation procedures.

Abstract: The MAC (Marker and Cell) discretization of fluid flow is analysed for the stationary Stokes equations. It is proved that the discrete approximations do in fact converge to the exact solutions of the flow equations. Estimates using mesh dependent norms analogous to the standard ${\bf H}^1 $ and $L^2 $ norms are given for the velocity and pressure, respectively.

Abstract: Fundamental Concepts Introduction Review of Electromagnetic Theory Classification of EM Problems Some Important Theorems Analytical Methods Introduction Separation of Variables Separation of Variables in Rectangular Coordinates Separation of Variables in Cylindrical Coordinates Separation of Variables in Spherical Coordinates Some Useful Orthogonal Functions Series Expansion Practical Applications Attenuation Due to Raindrops Concluding Remarks Finite Difference Methods Introduction Finite Difference Schemes Finite Differencing of Parabolic PDEs Finite Differencing of Hyperbolic PDEs Finite Differencing of Elliptic PDEs Accuracy and Stability of FD Solutions Practical Applications I - Guided Structures Practical Applications II - Wave Scattering (FDTD) Absorbing Boundary Conditions for FDTD Finite Differencing for Nonrectangular Systems Numerical Integration Concluding Remarks Variational Methods Introduction Operators in Linear Spaces Calculus of Variations Construction of Functionals from PDEs Rayleigh-Ritz Method Weighted Residual Method Eigenvalue Problems Practical Applications Concluding Remarks Moment Methods Introduction Integral Equations Green's Functions Applications I - Quasi-Static Problems Applications II - Scattering Problems Applications III- Radiation Problems Applications IV - EM Absorption in the Human Body Concluding Remarks Finite Element Method Introduction Solution of Laplace's Equation Solution of Poisson's Equation Solution of the Wave Equation Automatic Mesh Generation I - Rectangular Domains Automatic Mesh Generation II - Arbitrary Domains Bandwidth Reduction Higher Order Elements Three-Dimensional Elements Finite Element Methods for Exterior Problems Finite-Element Time-Domain Method Concluding Remarks Transmission-line-matrix Method Introduction Transmission-line Equations Solution of Diffusion Equation Solution of Wave Equations Inhomogeneous and Lossy Media in TLM Three-Dimensional TLM Mesh Error Sources and Correction Absorbing Boundary Conditions Concluding Remarks Monte Carlo Methods Introduction Generation of Random Numbers and Variables Evaluation of Error Numerical Integration Solution of Potential Problems Regional Monte Carlo Methods Time-Dependent Problems Concluding Remarks Method of Lines Introduction Solution of Laplace's Equation Solution of Wave Equation Time-Domain Solution Concluding Remarks References Problems APPENDICES Vector Relations Vector Identities Vector Theorems Orthogonal Coordinates Programming in MATLAB MATLAB Fundamentals Using MATLAB to Plot Programming with MATLAB Functions Solving Equations Programming Hints Other Useful MATLAB Commands Solution of Simultaneous Equations Elimination Methods Iterative Methods Matrix Inversion Eigenvalue Problems Answers to Odd-Numbered Problems

Abstract: SUMMARY A hybrid method for computing the flow of viscoelastic and second-order fluids is presented. It combines the features of the finite difference technique and the shooting method. The method is accurate because it uses central differences. Its convergence is at least superlinear. The method is applied to obtain the solutions to three problems of flow of Walters’ B fluid (a) flow near a stagnation point, (b) flow over a stretching sheet and (c) flow near a rotating disk. Numerical results reveal some new characteristics of flows which are not easy to demonstrate using the perturbation technique.

Abstract: Computations of electromagnetic fields are based either on differential equations or on integral equations. The differential equation approach using finite difference or finite element methods results in sparse matrices, which is an advantage, but has to cover large volumes, which is a disadvantage. The integral equation approach using the method of moments (MOM) limits the mesh to the surface of the object, which is an advantage, but results in full matrices, which is a disadvantage. It is noted that the ideal case would be to reduce the finite difference type equations close to the object surface and still preserve the sparsity of the matrices. The measured equation of invariance is a new concept in field computation capable of approaching this ideal situation. The mathematics and reasonings to reach a novel computational method based on this concept are presented. It is shown that the method is robust for both convex and concave objects, is much faster than the MOM, and uses a fraction of the memory. >

Abstract: The finite-difference method in the frequency domain is a powerful tool for analyzing arbitrarily shaped transmission-line discontinuities and junctions. An improved formulation based on Maxwell's equations in integral form is presented. It corresponds to the Helmholtz equation and reduces the numerical efforts in solving the large linear equation system considerably. The method is verified by comparison to previous work on microstrip. >

Abstract: Some of the most widely used algorithms for two-point boundary value ordinary differential equations, namely, finite-difference and collocation methods and standard multiple shooting, proceed by setting up and solving a structured system of linear equations. It is well known that the linear system can be set up efficiently in parallel; we show here that a structured orthogonal factorization technique can be used to solve this system, and hence the overall problem, in an efficient, parallel, and stable way.

Abstract: The use of digital filtering and spectrum estimation techniques for improving the efficiency of the FD-TD algorithm in solving eigenvalue problems is discussed. The great improvement of the efficiency of the method is demonstrated by means of both numerical and measurement results. In addition, several improvements to the present FD-TD method for eigenvalue analysis are presented. These include the analysis of open dielectric resonators and the extraction of the resonant frequencies from the FD-TD results. The result for the open dielectric resonator analysis is validated using measured data. >

Abstract: An approximation to a linear differential inclusion by means of N-stage single step discrete inclusions is presented, which is of second-order accuracy with respect to N. Approximations of this type with higher order of accuracy are shown not to exist, in general. The result is applied for a second-order discretization of control constrained optimal control problems and approximate feedback design making use of dynamic programming.

Abstract: This paper presents the derived Roe's flux difference splitting method for Payne's formulation of the macroscopic model. In recent years, this finite difference method has generated much interest and been used successfully in gas dynamics. Payne's equations are actually those of the isentropic gas having constant speed of sound and in homogeneous terms on the right hand side. Our motivation for deriving Roe's flux difference splitting algorithm for Payne's model and also highlighting the scalar equivalent of Roe's method (the Murman scheme) for the Lighthill-Whitham (LW) model stems from the reports of numerical simulation difficulties pertaining to two models. The numerical schemes were used on Payne's and LW models to simulate three traffic scenarios with satisfactory results. We were able to simulate the propagation of congested density upstream in one freeway bottleneck scenario at a finer discretization. Another scenario did not manage to obtain realistic results with Payne's model, we did.

Abstract: A standard parabolic equation (SPE) is used to approximate the Helmholtz equation for electromagnetic propagation in an inhomogeneous atmosphere. An implicit finite difference (IFD) scheme to solve the SPE is applied between the irregularly shaped ground and an altitude z=z/sub h/, below which all inhomogeneities of the medium are assumed localized. The boundary condition at z=z/sub h/ is obtained by matching the IFD solution to a surface Green's function (SGF) solution within the uniform region above z=z/sub h/. For ground slopes above about 1 degrees , the IFD implementation of the impedance boundary condition at the ground is shown to maintain the validity of the approximation only for vertically polarized waves. Predictions using this hybrid finite difference (FD)-SGF method agree well with results obtained using other computational methods. >

Abstract: A rigorous convergence result is presented for a finite difference scheme for the Navier-Stokes equations which uses vorticity boundary conditions The approximating scheme is based on the vorticity-stream function formulation of the Navier-Stokes equations The no-slip boundary condition is satisfied approximately by using a boundary condition of vorticity creation type Convergence with second-order accuracy in vorticity and velocity is established for general domains in two space dimensions Generalization to three space dimensions is also considered

Abstract: A computational study of shock-wave/boundary-layer interactions involving premixed combustible gases is presented. The analysis is carried out using a new fully implicit, total variation diminishing code that solves the fully coupled Reynolds-averaged Navier-Stokes equations and species continuity equations in an efficient manner. To accelerate the convergence of the basic iterative procedure, the code is combined with vector extrapolation methods. The chemical nonequilibrium processes are stimulated by means of a finite-rate chemistry model for hydrogen-air combustion

Abstract: A detailed formula is developed that can be used in both finite-volume (FV) and finite-difference (FD) methods for constructing freestream capturing metrics in space and time. It is shown that, considering an FV cell on the FD grid, the freestream capturing metrics in space and time can be constructed from the FD formulation. The approach is costly but guarantees the global conservation for an arbitrary motion of the grid.

Abstract: A simple technique is presented for the numerical solution of two-dimensional time-dependent flows, either laminar or turbulent, involving multiple free surfaces of arbitrary configuration The governing equations are the Reynolds equations for incompressible fluids with Boussinesq closure, the k- and ϵ-equations and an additional equation describing the fluid configuration This technique can potentially describe the propagation, deformation and overturning of pre-breaking waves and the mean flow, surface configuration and turbulence field after breaking The properties of the method are illustrated by several calculational examples The main parts of the algorithm are optimized for vector processing in a form suitable for installation in supercomputer facilities

Abstract: Based on approximation of the balance equation, finite difference schemes on triangular cell-centered grids are derived. A priori estimates and error analyses are provided. For certain regular triangulations, $\mathbb{O}(h^2 )$ convergence in the discrete $\mathbb{H}^1 $-norm is established. Also, finite difference schemes on triangular cell-centered grids with local refinement are derived with accuracy $\mathbb{O}(h^{1/2 + \alpha } )$, where $\alpha = 0$ for a simple symmetric scheme, $\alpha = 1$ for a nonsymmetric and a more accurate symmetric scheme, and $\alpha = \frac{3}{2}$ for a more accurate nonsymmetric scheme.Certain algebraic properties of the corresponding matrices of the derived finite difference schemes are verified, thus allowing the recently proposed algebraic theory for the Bramble–Ewing–Pasciak–Schatz (BEPS) and Fact Adaptive Composite (FAC) two-grid preconditioners to apply.Numerical experiments that demonstrate the accuracy of the difference schemes and the fast convergence of the two...

Abstract: We analyze a Crank-Nicolson-type finite difference scheme for the Kuramoto-Sivashinsky equation in one space dimension with periodic boundary conditions. We discuss linearizations of the scheme and derive second-order error estimates.