Abstract: We propose and study a numerical method for time discretization of linear and semilinear integro-partial differential equations that are intermediate between diffusion and wave equations, or are subdiffusive. The method uses convolution quadrature based on the second-order backward differentiation formula. Second-order error bounds of the time discretization and regularity estimates for the solution are shown in a unified way under weak assumptions on the data in a Banach space framework. Numerical experiments illustrate the theoretical results.

Abstract: Heat Conduction Fundamentals.- Fourier Law.- Mass and Energy Balance Equations.- The Reduction of Transient Heat Conduction Equations and Boundary Conditions.- Substituting Heat Conduction Equation by Two-Equations System.- Variable Change.- Exercises. Solving Heat Conduction Problems.- Heat Transfer Fundamentals.- Two-Dimensional Steady-State Heat Conduction. Analytical Solutions.- Analytical Approximation Methods. Integral Heat Balance Method.- Two-Dimensional Steady-State Heat Conduction. Graphical Method.- Two-Dimensional Steady-State Problems. The Shape Coefficient.- Solving Steady-State Heat Conduction Problems by Means of Numerical Methods.- Finite Element Balance Method and Boundary Element Method.- Transient Heat Exchange between a Body with Lumped Thermal Capacity and Its Surroundings.- Transient Heat Conduction in Half-Space.- Transient Heat Conduction in Simple-Shape Elements.- Superposition Method in One-Dimensional Transient Heat Conduction Problems.- Transient Heat Conduction in a Semi-Infinite Body. The Inverse Problem.- Inverse Transient Heat Conduction Problems.- Multidimensional Problems. The Superposition Method.- Approximate Analytical Methods for Solving Transient Heat Conduction Problems.- Finite Difference Method.- Solving Transient Heat Conduction Problems by Means of Finite Element Method (FEM).- Numerical-Analytical Methods.- Solving Inverse Heat Conduction Problems by Means of Numerical Methods.- Heat Sources.- Melting and Solidification (Freezing).

Abstract: Heterogeneous finite-difference (FD) modeling assumes that the boundary conditions of the elastic wavefield between material discontinuities are implicitly fulfilled by the distribution of the elastic parameters on the numerical grid. It is widely applied to weak elastic contrasts between geologic formations inside the earth. We test the accuracy at the free surface of the earth. The accuracy for modeling Rayleigh waves using the conventional standard staggered-grid (SSG) and the rotated staggered grid (RSG) is investigated. The accuracy tests reveal that one cannot rely on conventional numerical dispersion discretization criteria. A higher sampling is necessary to obtain acceptable accuracy. In the case of planar free surfaces aligned with the grid, 15 to 30 grid points per minimum wavelength of the Rayleigh wave are required. The widely used explicit boundary condition, the so-called image method, produces similar accuracy and requires approximately half the sampling of the wavefield compared to heterogeneous free-surface modeling. For a free-surface not aligned with the grid (surface topography), the error increases significantly and varies with the dip angle of the interface. For an irregular interface, the RSG scheme is more accurate than the SSG scheme. The RSG scheme, however, requires 60 grid points per minimum wavelength to achieve good accuracy for all dip angles. The high computation requirements for 3D simulations on such fine grids limit the application of heterogenous modeling in the presence of complex surface topography.

Abstract: The probability density evolution method (PDEM) for dynamic responses analysis of non-linear stochastic structures is proposed. In the method, the dynamic response of non-linear stochastic structures is firstly expressed in a formal solution, which is a function of the random parameters. In this sense, the dynamic responses are mutually uncoupled. A state equation is then constructed in the augmented state space. Based on the principle of preservation of probability, a one-dimensional partial differential equation in terms of the joint probability density function is set up. The numerical solving algorithm, where the Newmark-Beta time-integration algorithm and the finite difference method with Lax–Wendroff difference scheme are brought together, is studied. In the numerical examples, free vibration of a single-degree-of-freedom non-linear conservative system and dynamic responses of an 8-storey shear structure with bilinear hysteretic restoring forces, subjected to harmonic excitation and seismic excitation, respectively, are investigated. The investigations indicate that the probability density functions of dynamic responses of non-linear stochastic structures are usually irregular and far from the well-known distribution types. They exhibit obvious evolution characteristics. The comparisons with the analytical solution and Monte Carlo simulation method demonstrate that the proposed PDEM is of fair accuracy and efficiency. Copyright © 2005 John Wiley & Sons, Ltd.

Abstract: Numerical Convergence of the MPFA O-Method for General Quadrilateral Grids in Two and Three Dimensions.- Differential Complexes and Stability of Finite Element Methods I. The de Rham Complex.- Defferential Complexes and Stability of Finite Element Methods II: The Elasticity Complex.- On the Role of Involutions in the Discontinuous Galerkin Discretization of Maxwell and Magnetohydrodynamic Systems.- Principles of Mimetic Discretizations of Differential Operators.- Compatible Discretizations for Eigenvalue Problems.- Conjugated Bubnov-Galerkin Infinite Element for Maxwell Equations.- Covolume Discretization of Differential Forms.- Mimetic Reconstruction of Vectors.- A Cell-Centered Finite Difference Method on Quadrilaterals.- Development and Application of Compatible Discretizations of Maxwell's Equations.

Abstract: The electromagnetic fields surrounding a thin, subseabed resistive disk in response to a deep-towed, time-harmonic electric dipole antenna are investigated using a newly developed 3D Cartesian, staggered-grid modeling algorithm. We demonstrate that finite-difference and finite-volume methods for solving the governing curl-curl equation yield identical, complex-symmetric coefficient matrices for the resulting N×N linear system of equations. However, the finite-volume approach has an advantage in that it naturally admits quadrature integration methods for accurate representation of highly compact or exponentially varying source terms constituting the right side of the resulting linear system of equations. This linear system is solved using a coupled two-term recurrence, quasi-minimal residual algorithm that doesnot require explicit storage of the coefficient matrix, thus reducing storage costs from 22N to 10N complex, double-precision words with no decrease in computational performance. The disk model serve...

Abstract: SUMMARY We present a seismic waveform inversion methodology based on the Gauss–Newton method from pre-stack seismic data. The inversion employs a staggered-grid finite difference solution of the 2-D elastic wave equation in the time domain, allowing accurate simulation of all possible waves in elastic media. The partial derivatives for the Gauss–Newton method are obtained from the differential equation of the wave equation in terms of model parameters. The resulting wave equation and virtual sources from the reciprocity principle allow us to apply the Gauss–Newton method to seismic waveform inversion. The partial derivative wavefields are explicitly computed by convolution of forward wavefields propagated from each source with reciprocal wavefields from each receiver. The Gauss–Newton method for seismic waveform inversion was proposed in the 1980s but has rarely been studied. Extensive computational and memory requirements have been principal difficulties which are addressed in this work. We used different sizes of grids for the inversion, temporal windowing, approximation of virtual sources, and parallelizing computations. With numerical experiments, we show that the Gauss–Newton method has significantly higher resolving power and convergence rate over the gradient method, and demonstrate potential applications to real seismic data.

Abstract: The response of a structural system to white noise excitation (deltacorrelated) constitutes a Markov vector process whose transitional probability density function (TPDF) is governed by both the forward Fokker-Planck and backward Kolmogorov equations. Numerical solution of these equations by finite element and finite difference methods for dynamical systems of engineering interest has been hindered by the problem of dimensionality. In this paper numerical solution of the stationary and transient form of the Fokker-Planck (FP) equation corresponding to two state nonlinear systems is obtained by standard sequential finite element method (FEM) using C0 shape function and Crank-Nicholson time integration scheme. The method is applied to Van-der-Pol and Duffing oscillators providing good agreement between results obtained by it and exact results. An extension of the finite difference discretization scheme developed by Spencer, Bergman and Wojtkiewicz is also presented. This paper presents an extension of the finite difference method for the solution of FP equation up to four dimensions. The difficulties associated in extending these methods to higher dimensional systems are discussed.

Abstract: This paper presents an extension to the work presented in Part I of this series of two articles to the transient case. Emphasis is placed on the development of a new model for heat flow in a double U-shape vertical borehole heat exchanger and its thermodynamic interactions with surrounding soil mass. The discretization of the spatial-temporal domain of the heat pipe model is done by the use of the space–time finite element technique in conjunction with the Petrov–Galerkin method and the finite difference method. The paper shows that the proposed model and the choice of the discretization technique, in addition to the utilization of a sequential numerical algorithm for solving the resulting system of non-linear equations, have contributed in reducing significantly the required number of finite elements necessary for describing geothermal heating systems. Details of the mathematical derivations and comparison to experimental data are presented. Copyright © 2006 John Wiley & Sons, Ltd.

Abstract: A meshless numerical model is developed for analyzing transient heat conduction in non-homogeneous functionally graded materials (FGM), which has a continuously functionally graded thermal conductivity parameter First, the analog equation method is used to transform the original non-homogeneous problem into an equivalent homogeneous one at any given time so that a simpler fundamental solution can be employed to take the place of the one related to the original problem Next, the approximate particular and homogeneous solutions are constructed using radial basis functions and virtual boundary collocation method, respectively Finally, by enforcing satisfaction of the governing equation and boundary conditions at collocation points of the original problem, in which the time domain is discretized using the finite difference method, a linear algebraic system is obtained from which the unknown fictitious sources and interpolation coefficients can be determined Further, the temperature at any point can be easily computed using the results of fictitious sources and interpolation coefficients The accuracy of the proposed method is assessed through two numerical examples

Abstract: In problems with interfaces, the unknown or its derivatives may have jump discontinuities. Finite difference methods, including the method of A. Mayo and the immersed interface method of R. LeVeque and Z. Li, maintain accuracy by adding corrections, found from the jumps, to the difference operator at grid points near the interface and by modifying the operator if necessary. It has long been observed that the solution can be computed with uniform O.h 2 / accuracy even if the truncation error is O.h/ at the interface, while O.h 2 / in the interior. We prove this fact for a class of static interface problems of elliptic type using discrete analogues of estimates for elliptic equations. Moreover, we show that the gradient is uniformly accurate to O.h 2 log.1=h//. Various implications are discussed, including the accuracy of these methods for steady fluid flow governed by the Stokes equations. Two-fluid problems can be handled by first solving an integral equation for an unknown jump. Numerical examples are presented which confirm the analytical conclusions, although the observed error in the gradient is O.h 2 /.

Abstract: The Finite Difference Method (FDM), within the framework of the Meshless Local PetrovGalerkin (MLPG) approach, is proposed in this paper for solving solid mechanics problems. A “mixed” interpolation scheme is adopted in the present implementation: the displacements, displacement gradients, and stresses are interpolated independently using identical MLS shape functions. The system of algebraic equations for the problem is obtained by enforcing the momentum balance laws at the nodal points. The divergence of the stress tensor is established through the generalized finite difference method, using the scattered nodal values and a truncated Taylor expansion. The traction boundary conditions are imposed in the stress equations directly, using a local coordinate system. Numerical examples show that the proposed MLPG mixed finite difference method is both accurate and efficient, and stable. keyword: Meshless method, Finite difference method, MLPG

Abstract: In order to solve the problem of truncating the open boundaries of cylindrical waveguides used in the simulation of high-power microwave (HPM) sources, this paper studies the convolutional perfectly matched layer (CPML) in the cylindrical coordinate system. The electromagnetic field's finite-difference time-domain (FDTD) equations and the expressions of axis boundary conditions are presented. Numerical experiments are conducted to validate the equations and axis boundary conditions. The performance of CPML is simulated when it is used to truncate the cylindrical waveguide excited by the sources with different frequencies and modes in the two-and-a-half-dimensional (2.5-D) problems. Numerical results show that the maximum relative error is less than -95 dB, and demonstrate that the property of CPML is much better than that of the Mur-type absorbing boundary condition when they are used to truncate the open boundaries of waveguides. The CPML is especially suitable for truncating the open boundaries of the dispersive waveguide devices in the simulation of HPM sources

Abstract: A linearized implicit finite difference method to obtain numerical solution of the one-dimensional regularized long-wave (RLW) equation is presented. The performance and the accuracy of the method are illustrated by solving three test examples of the problem: a single solitary wave, two positive solitary waves interaction, and an undular bore. The obtained results are presented and compared with earlier work.

Abstract: [1] High-frequency seismograms mainly consist of incoherently scattered waves. Although their phases are more or less random, their envelopes show smooth and stable variations depending on frequency and distance. Envelope modeling can thus be used to infer stochastic parameters of the heterogeneous Earth medium. Radiative transfer theory (RTT) describes energy transport through a random heterogeneous medium neglecting phase information and has been frequently used to simulate observed mean square (MS) envelopes of high-frequency waves. The radiative transfer equations can be numerically solved by Monte Carlo simulations. So far, mostly isotropic scattering and acoustic approximations have been used. Here we present an extension of the Monte Carlo method to the full elastic case including P, S, and conversion scattering where the single scattering events are described by angular-dependent scattering coefficients in random media which follow from the Born approximation. In order to validate the method, the simulated envelopes are compared to average envelopes obtained by full waveform modeling with a finite difference method in two-dimensional random media with Gaussian and exponential correlation functions. Envelope shapes agree remarkably well for both short and long lapse times and for a broad range of scattering parameters. We conclude that the use of Born scattering coefficients in RTT does not pose severe limits on its validity range. Even in the strong forward scattering regime, envelope broadening and peak amplitude delays can be successfully modeled if one includes the wandering effect as obtained from the parabolic wave equation and Markov approximation into RTT.

Abstract: This paper describes the implementation and successful validation of a new staggered-grid, finite-difference algorithm for the numerical simulation of frequency-domain electromagnetic borehole measurements. The algorithm is basedonacoupledscalar-vectorpotentialformulationforarbitrary 3D inhomogeneous electrically anisotropic media. We approximate the second-order partial differential equations for the coupled scalar-vector potentials with central finite differences on both Yee’s staggered and standard grids. Thediscretizationofthepartialdifferentialequationsandthe enforcement of the appropriate boundary conditions yields a complex linear system of equations that we solve iteratively using the biconjugate gradient method with preconditioning. Theaccuracyandefficiencyofthealgorithmisassessedwith examples of multicomponent-borehole electromagnetic-induction measurements acquired in homogeneous, 1D anisotropic,2Disotropic,and3Danisotropicrockformations.The simulation examples consider vertical and deviated wells with and without borehole and mud-filtrate invasion regions. Simulation results obtained with the scalar-vector coupled potentialformulationfavorablycompareinaccuracywithresults obtained with 1D, 2D, and 3D benchmarking codes in the dc to megahertz frequency range for large contrasts of electricalconductivity.Ournumericalexercisesindicatethat the coupled scalar-vector potential equations provide a generalandconsistentalgorithmicformulationtosimulateborehole electromagnetic measurements from dc to megahertz in the presence of large conductivity contrasts, dipping wells, electrically anisotropic media, and geometrically complex modelsofelectricalconductivity.