Abstract: In this review paper, the finite difference methods (FDMs) for the fractional differential equations are displayed. The considered equations mainly include the fractional kinetic equations of diffusion or dispersion with time, space and time-space derivatives. In some way, these numerical methods have similar form as the case for classical equations, some of which can be seen as the generalizations of the FDMs for the typical differential equations. And the classical tools, such as the von Neumann analysis method, the energy method and the Fourier method are extended to numerical methods for fractional differential equations accordingly. At the same time, the techniques for improving the accuracy and reducing the computation and storage are also introduced.

Abstract: Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Because of the nonlocal property of fractional differential operators, numerical methods for space-fractional diffusion equations generate complicated dense or full coefficient matrices. Consequently, these numerical methods were traditionally solved by Gaussian elimination, which requires computational work of $O(N^3)$ per time step and $O(N^2)$ of memory, where $N$ is the number of spatial grid points in the discretization. The significant computational work and memory requirement of the numerical methods impose a serious challenge for the numerical simulation of two- and especially three-dimensional space-fractional diffusion equations. We develop a fast and yet accurate solution method for the implicit finite difference discretization of space-fractional diffusion equations in two space dimensions by carefully analyzing the structure of the coefficient ma...

Abstract: The analysis of adhesively bonded joints started in 1938 with the closed-form model of Volkersen. The equilibrium equation of a single lap joint led to a simple governing differential equation with a simple algebraic equation. However, if there is yielding of the adhesive and/or the adherends and substantial peeling is present, a more complex model is necessary. The more complete is an analysis, the more complicated it becomes and the more difficult it is to obtain a simple and effective solution. The finite element (FE) method, the boundary element (BE) method and the finite difference (FD) method are the three major numerical methods for solving differential equations in science and engineering. These methods have also been applied to adhesive joints, especially the FE method. This book deals with the most recent numerical modelling of adhesive joints. Advances in damage mechanics and extended finite element method are described in the context of the FE method with examples of application. The classical continuum mechanics and fracture mechanics approach are also introduced. The BE method and the FD method are also discussed with indication of the cases they are most adapted to. There is not at the moment a numerical technique that can solve any problem and the analyst needs to be aware of the limitations involved in each case.

Abstract: Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid1940s. In a 1949 letter von Neumann wrote “the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both.” The “greater whole” is viewed today as scientific computation: over the past sixty years, scientific computation has emerged as the most versatile tool to complement theory and experiments, and numerical methods for solving PDEs are at the heart of many of today’s advanced scientific computations. Numerical solutions found their way from financial models on Wall Street to traffic models on Main Street. Here we provide a bird’s eye view on the development of these numerical methods with a particular emphasis on nonlinear PDEs.

Abstract: We analyze finite difference methods for the Gross-Pitaevskii equation with an angular momentum rotation term in two and three dimensions and obtain the optimal convergence rate, for the conservative Crank-Nicolson finite difference (CNFD) method and semi-implicit finite difference (SIFD) method, at the order of O(h2 + τ2) in the l2-norm and discrete H1-norm with time step τ and mesh size h. Besides the standard techniques of the energy method, the key technique in the analysis for the SIFD method is to use the mathematical induction, and resp., for the CNFD method is to obtain a priori bound of the numerical solution in the l∞-norm by using the inverse inequality and the l2-norm error estimate. In addition, for the SIFD method, we also derive error bounds on the errors between the mass and energy in the discretized level and their corresponding continuous counterparts, respectively, which are at the same order of the convergence rate as that of the numerical solution itself. Finally, numerical results are reported to confirm our error estimates of the numerical methods.

Abstract: We establish uniform error estimates of finite difference methods for the nonlinear Schrodinger equation (NLS) perturbed by the wave operator (NLSW) with a perturbation strength described by a dimensionless parameter $\varepsilon$ ($\varepsilon\in(0,1]$). When $\varepsilon\to0^+$, NLSW collapses to the standard NLS. In the small perturbation parameter regime, i.e., $0<\varepsilon\ll1$, the solution of NLSW is perturbed from that of NLS with a function oscillating in time with $O(\varepsilon^2)$-wavelength at $O(\varepsilon^4)$ and $O(\varepsilon^2)$ amplitudes for well-prepared and ill-prepared initial data, respectively. This high oscillation of the solution in time brings significant difficulties in establishing error estimates uniformly in $\varepsilon$ of the standard finite difference methods for NLSW, such as the conservative Crank-Nicolson finite difference (CNFD) method, and the semi-implicit finite difference (SIFD) method. We obtain error bounds uniformly in $\varepsilon$, at the order of $O(h^2+\tau)$ and $O(h^2+\tau^{2/3})$ with time step $\tau$ and mesh size $h$ for well-prepared and ill-prepared initial data, respectively, for both CNFD and SIFD in the $l^2$-norm and discrete semi-$H^1$ norm. Our error bounds are valid for general nonlinearity in NLSW and for one, two, and three dimensions. To derive these uniform error bounds, we combine $\varepsilon$-dependent error estimates of NLSW, $\varepsilon$-dependent error bounds between the numerical approximate solutions of NLSW and the solution of NLS, together with error bounds between the solutions of NLSW and NLS. Other key techniques in the analysis include the energy method, cut-off of the nonlinearity, and a posterior bound of the numerical solutions by using the inverse inequality and discrete semi-$H^1$ norm estimate. Finally, numerical results are reported to confirm our error estimates of the numerical methods and show that the convergence rates are sharp in the respective parameter regimes.

Abstract: SUMMARY Surface topography has been considered a difficult task for seismic wave numerical modelling by the finite-difference method (FDM) because the most popular staggered finite-difference scheme requires a rectilinear grid. Even though there are numerous collocated grid schemes in other computational fields that could be used to solve the first-order hyperbolic equations, the lack of a stable free-surface boundary condition implementation for curvilinear grids also obstructs the adoption of curvilinear grids in seismic wave FDM modelling. In this study, we use generalized curvilinear grids that can fit the surface topography to discretize the computational domain and describe the implementation of a collocated grid finite-difference scheme, a higher order MacCormack scheme, to solve the first-order hyperbolic velocity-stress equations on the curvilinear grid. To achieve a sufficiently accurate and stable free-surface boundary condition implementation on the curvilinear grids, we propose the traction image method that antisymmetrically images the traction components instead of the stress components to the ghost points above the free surface. Since the velocity derivatives at the free surface are provided by the free-surface condition, we use a compact scheme to compute the velocity derivatives near the free surface and avoid the use of velocity values on the ghost points. Numerical tests verify that using the curvilinear grid, the collocated finite-difference scheme and the traction image technique can simulate seismic wave propagation in the presence of surface topography with sufficient accuracy.

Abstract: A block-centered finite difference scheme is introduced to solve the nonlinear Darcy--Forchheimer equation, in which the velocity and pressure can be approximated simultaneously. The second-order error estimates for both pressure and velocity are established on a nonuniform rectangular grid. Numerical experiments using the scheme show the consistency of the convergence rates of our method with the theoretical analysis.

Abstract: In this paper, the continuous genetic algorithm is applied for the solution of singular two-point boundary value problems, where smooth solution curves are used throughout the evolution of the algorithm to obtain the required nodal values. The proposed technique might be considered as a variation of the finite difference method in the sense that each of the derivatives is replaced by an appropriate difference quotient approximation. This novel approach possesses main advantages; it can be applied without any limitation on the nature of the problem, the type of singularity, and the number of mesh points. Numerical examples are included to demonstrate the accuracy, applicability, and generality of the presented technique. The results reveal that the algorithm is very effective, straightforward, and simple.

Abstract: Background in Mechanics. Physical Objects. System of Fluid-Dynamics Equations for Homogeneous Isotropic Incompressible Flows. Two-Dimensional System of Shallow-Water Equations (2-D SSWE). Physical Meanings of Various Terms in 2-D SSWE. Various Forms of 2-D SSWE. Properties of 2-D SSWE. Conceptual Mechanical Behavior. Dimensional Analysis of 2-D SSWE. Basic Mathematics for Systems of First-Order Quasilinear Hyperbolic Equations. Geometric Theory of Characteristics. Riemann Invariants. Theory of Nonlinear Wave Propagation. Properties of the Solutions of 2-D SSWE. Initial and Boundary Conditions for Well-Posed Problems. Behavior of Solutions. Discontinuous Solutions of SSWE. Isentropic Flow Simulation of SSWE and its Limitations. Discontinuous Solutions of 1-D First-Order Hyperbolic Systems. Introduction to 2-D Discontinuous Solutions. Mathematical Conditions of Shock Waves for 2-D SSWE. Preliminary Review of Finite Difference Methods. General Description. Basic Performance of a Difference Scheme. Basic Difference Schemes for First-Order Hyperbolic Systems in One Space Dimension. FDMs for the Computation of 1-D Unsteady Open Flows. Difference Schemes for 2-D SSWE. FDMs for the Solution of 2-D SSWE in Nonconservative Form. FDMs for the Solution of 2-D SSWE in Conservative Form. Fractional-Step Methods and Splitting-Up Algorithms. Fractional-Step Difference Schemes for 2-D Unsteady Flow Computations. FDMs for Curvilinear Meshes. Finite Volume Method (FVM). Numerical Solutions Using Finite Element Methods. Related Principles in Variational Calculus. Piecewise Approximation of Plane Problems and Convergence of FEM Solutions. FEM for 2-D Unsteady Open Flows. Several Classes of Special FEMs. Techniques for the Implementation of Algorithms. Computational Mesh. Classical Techniques for Improving Computational Stability and Accuracy. New Developments of Difference Schemes for 2-D First-Order Hyperbolic Systems of Equations. General Description. Two-Dimensional Methods of Characteristics. Characteristic-Based Splitting. Riemann Approach. Approximate Factorization of Implicit Schemes. FCT Algorithms and TVD Schemes. Square Conservation and Energy Conservation Schemes. Stability Analysis and Boundary Procedures. Mathematical Definitions of Stability for Difference Schemes. Von Neumann Linear Stability Analysis. Nonlinear Instability. Boundary Procedures and Their Influence on Numerical Solutions. Stability Theory for Mixed Problems. Concluding Remarks. Requirements for an Ideal Finite-Difference Scheme. Comparison of Performance, Merits and Drawbacks Between FDM and FEM. Brief Introduction to Other Algorithms. Towards a Truly 2-D Algorithm. Index.

Abstract: This paper describes a new stochastic finite difference time domain (S-FDTD) method for calculating the variance in the electromagnetic fields caused by variability or uncertainty in the electrical properties of the materials in the model. Details of the 1D derivation using truncated Taylor series approximations are given. The S-FDTD analysis is then compared to Monte Carlo analysis for a 1D bioelectromagnetic example. The accuracy of the method is controlled by the approximations for the cross correlations of the fields and electrical properties. In this paper, we are able to bound the variance using cross correlations of 1 (overestimate) and the reflection coefficient (underestimate).

Abstract: An one-step arbitrary-order leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is presented. Its unconditional stability is analytically proven and numerically verified. Its numerical properties are shown to be identical to those of the conventional ADI-FDTD and LOD-FDTD methods. However, its computational expenditure is found to be the lowest. In other words, in comparison with the ADI-FDTD and LOD-FDTD methods, the one-step arbitrary-order leap-frog ADI-FDTD method retains identical numerical modeling accuracy but with higher computational efficiency.

Abstract: SUMMARY To simulate seismic wave propagation in the spherical Earth, the Earth’s curvature has to be taken into account. This can be done by solving the seismic wave equation in spherical coordinates by numerical methods. In this paper, we use an optimized, collocated-grid finite-difference scheme to solve the anisotropic velocity–stress equation in spherical coordinates. To increase the efficiency of the finite-difference algorithm, we use a non-uniform grid to discretize the computational domain. The grid varies continuously with smaller spacing in low velocity layers and thin layer regions and with larger spacing otherwise. We use stress-image setting to implement the free surface boundary condition on the stress components. To implement the free surface boundary condition on the velocity components, we use a compact scheme near the surface. If strong velocity gradient exists near the surface, a lower-order scheme is used to calculate velocity difference to stabilize the calculation. The computational domain is surrounded by complex-frequency shifted perfectly matched layers implemented through auxiliary differential equations (ADE CFS-PML) in a local Cartesian coordinate. We compare the simulation results with the results from the normal mode method in the isotropic and anisotropic models and verify the accuracy of the finite-difference method.

Abstract: In this paper, we develop the Crank-Nicolson nite dierence method (C-N-FDM) to solve the linear time-fractional diusion equation, for- mulated with Caputo's fractional derivative. Special attention is given to study the stability of the proposed method which is introduced by means of a recently proposed procedure akin to the standard Von-Neumann stable analysis. Some numerical examples are presented and the behavior of the solution is examined to verify stability of the proposed method. It is found that the C-N-FDM is applicable, simple and ecient for such problems.

Abstract: The complex natural frequencies for linear free vibrations and bifurcation and chaos for forced nonlinear vibration of axially moving viscoelastic plate are investigated in this paper. The governing partial differential equation of out-of-plane motion of the plate is derived by Newton’s second law. The finite difference method in spatial field is applied to the differential equation to study the instability due to flutter and divergence. The finite difference method in both spatial and temporal field is used in the analysis of a nonlinear partial differential equation to detect bifurcations and chaos of a nonlinear forced vibration of the system. Numerical results show that, with the increasing axially moving speed, the increasing excitation amplitude, and the decreasing viscosity coefficient, the equilibrium loses its stability and bifurcates into periodic motion, and then the periodic motion becomes chaotic motion by period-doubling bifurcation.

Abstract: In this paper, an unconditionally stable finite-difference time-domain algorithm is presented for analysis of lossy nonuniform multiconductor transmission line (MTL). The results of the proposed algorithm are confirmed by those of the Leap-Frog algorithm. The unconditionally stable algorithm alone cannot decrease the time consumption, especially when the MTL is excited by modulated signals. The complex phasor transformation approach is used to separate the modulating signal from the carrier signal when an MTL is excited by a modulated signal. A new set of equations is generated using this approach. The new equations are based on the modulating signal, and the carrier signal is eliminated in these equations. The derived equations are solved by Leap-Frog algorithm, and the results confirm the accuracy of these equations. Finally, the proposed equations are solved by the unconditionally stable algorithm at several temporal step sizes. The results indicate that this method has much lower CPU time compared with the conventional method and also has a very good accuracy.

Abstract: Groundwater contamination is a severe problem in many parts of the world including India. The complex problem of groundwater flow and contaminant transport is studied generally by solving the governing equations of flow and transport using numerical models such as finite difference method (FDM) or finite element method (FEM). Meshfree (MFree) method is an alternative numerical approach to solve these governing equations in simple and accurate manner. MFree method does not require any grid and only makes the use of a set of scattered collocation points, regardless of the connectivity information between them. Kansa (1990) [9] developed a multi-quadratic (MQ) based MFree method for the solution of partial differential equations. Based on the Kansa’s method, the present study proposes a MFree point collocation method (PCM) with multi-quadric radial basis function (MQ-RBF) for the two-dimensional coupled groundwater flow and transport simulation in unconfined conditions. The accuracy of the developed model is verified with available analytical solutions in literature. The coupled model developed is further applied to a field problem to compute the groundwater head and concentration distribution and the results are compared with available finite element based simulation. The outcomes of the model results showed the applicability of the present approach.

Abstract: In this work chaotic dynamics of continuous mechanical systems such as flexible plates and shallow shells is studied Namely, a wide class of the mentioned objects is analyzed including flexible plates and cylinder-like panels of infinite length, rectangular spherical and cylindrical shells, closed cylindrical shells, axially symmetric plates, as well as spherical and conical shells The considered problems are solved by the Bubnov–Galerkin and higher approximation Ritz methods Convergence and validation of those methods are studied The Cauchy problems are solved mainly by the fourth Runge-Kutta method, although all variants of the Runge-Kutta methods are considered New scenarios of transition from regular to chaotic orbits are detected, analyzed and discussed First part of the paper is devoted to the validation of results obtained This is why the same infinite length problem is reduced to that of a finite dimension through the FDM (Finite Difference Method) with the approximation order of O ( c 2 ), BGM (Bubnov–Galerkin Method) or RM (Ritz Method) with higher approximations We pay attention not only to convergence of the mentioned methods regarding the number of partitions of the interval [0, 1] in the FDM or regarding the number of terms in the series applied either in the BGM or RM methods, but we also compare the results obtained via the mentioned different approaches Furthermore, a so called practical convergence of different Runge-Kutta type methods are tested starting from the second and ending with the eighth order Second part of the work is devoted to a study of routes to chaos in the so far mentioned mechanical objects For this purpose the so-called “dynamical charts” are constructed versus control parameters { q 0 , ω p }, where q 0 denotes the loading amplitude, and ω p is the loading frequency The charts are constructed through analyses of frequency power spectra and the largest Lyapunov exponent (LE) Analysis of the mentioned charts indicates clearly that different routes to chaos exist and allow us to control the objects being investigated In some cases we detect the classical Feigenbaum scenario and we compute also the Feigenbaum constant This scenario accompanied all problems which we studied In addition, we detect and illustrate novel scenarios of transition from regularity into chaos including the Ruelle–Takens–Newhouse–Feigenbaum scenario, and the so called modified Pomeau–Manneville scenario Third part of the paper is devoted to analysis of the Lyapunov exponents Namely, while investigating evolutions of vibration regimes of a shell associated with an increase of excitation amplitude q 0 phase transitions chaos–hyper chaos as well as chaos-hyper chaos–hyper–hyper chaos dynamics are illustrated and studied Furthermore, for all investigated plates and shells the Sharkovskiy windows of periodicity are detected In particular, a space-temporal chaos/turbulence is studied

Abstract: This work presents a structural optimization aided design methodology for composite laminated plates subject to fluid-structure interaction. The goal of the optimization procedure is to increase the flutter speed onset through the maximization of natural frequencies related to the vibration modes involved in the phenomenon. The aeroelastic stability analysis is performed using ZAERO software system, which includes ZONA 6 unsteady lifting surface method. The finite element method is applied to solve the structural model equilibrium equations, the eigenvalues sensitivities with respect to design variables are calculated analytically, and sequential linear programming is applied. The maximization is accomplished using two methods; the first method uses an aeroelastic analysis to determine which eigenmode causes the flutter onset, and its eigenvalue is then maximized. In the second method, a forward finite difference method is applied and the flutter speed sensitivities with respect to the eigenvalues are calculated. This sensitivity is used to guide the optimization process. Finally, a topology optimization problem is formulated to reduce the plate mass under a minimum flutter velocity constraint, using density distribution as the design variable.

Abstract: We study the stability properties of, and the phase error present in, several higher-order (in space) staggered finite difference schemes for Maxwell’s equations coupled with a Debye or Lorentz polarization model. We present a novel expansion of the symbol of finite difference approximations, of arbitrary (even) order, of the first-order spatial derivative operator. This alternative representation allows the derivation of a concise formula for the numerical dispersion relation for all (even-) order schemes applied to each model, including the limiting (infinite-order) case. We further derive a closed-form analytical stability condition for these schemes as a function of the order of the method. Using representative numerical values for the physical parameters, we validate the stability criterion while quantifying numerical dissipation. Lastly, we demonstrate the effect that the spatial discretization order, and the corresponding stability constraint, has on the dispersion error.

Abstract: In this chapter, we initially give an introduction to methods for computing derivatives and partial derivatives using discrete differential operators and discuss the connection to Taylor series.