Abstract: A family of inexpensive discretization schemes for diffusion problems on unstructured polygonal and polyhedral meshes is introduced. The material properties are described by a full tensor. The theoretical results are confirmed with numerical experiments.

Abstract: We present a finite difference method for solving parabolic partial integro-differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Levy process or, more generally, a time-inhomogeneous jump-diffusion process. We discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the Levy measure. We propose an explicit-implicit finite difference scheme which can be used to price European and barrier options in such models. We study stability and convergence of the scheme proposed and, under additional conditions, provide estimates on the rate of convergence. Numerical tests are performed with smooth and nonsmooth initial conditions.

Abstract: The stability and convergence properties of the mimetic finite difference method for diffusion-type problems on polyhedral meshes are analyzed. The optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved.

Abstract: This paper surveys several topics related to the observation and control of wave propagation phenomena modeled by finite difference methods. The main focus is on the property of observability, corresponding to the question of whether the total energy of solutions can be estimated from partial measurements on a subregion of the domain or boundary. The mathematically equivalent property of controllability corresponds to the question of whether wave propagation behavior can be controlled using forcing terms on that subregion, as is often desired in engineering applications. Observability/controllability of the continuous wave equation is well understood for the scalar linear constant coefficient case that is the focus of this paper. However, when the wave equation is discretized by finite difference methods, the control for the discretized model does not necessarily yield a good approximation to the control for the original continuous problem. In other words, the classical convergence (consistency + stability) property of a numerical scheme does not suffice to guarantee its suitability for providing good approximations to the controls that might be needed in applications. Observability/controllability may be lost under numerical discretization as the mesh size tends to zero due to the existence of high-frequency spurious solutions for which the group velocity vanishes. This phenomenon is analyzed and several remedies are suggested, including filtering, Tychonoff regularization, multigrid methods, and mixed finite element methods. We also briefly discuss these issues for the heat, beam, and Schrodinger equations to illustrate that diffusive and dispersive effects may help to retain the observability/controllability properties at the discrete level. We conclude with a list of open problems and future subjects for research.

Abstract: The spontaneously propagating shear crack on a frictional interface has proven to be a useful idealization of a natural earthquake. The corresponding boundary value problems are nonlinear and usually require computationally intensive numerical methods for their solution. Assessing the convergence and accuracy of the numerical methods is challenging, as we lack appropriate analytical solutions for comparison. As a complement to other methods of assessment, we compare solutions obtained by two independent numerical methods, a finite difference method and a boundary integral (BI) method. The finite difference implementation, called DFM, uses a traction-at-split-node formulation of the fault discontinuity. The BI implementation employs spectral representation of the stress transfer functional. The three-dimensional (3-D) test problem involves spontaneous rupture spreading on a planar interface governed by linear slip-weakening friction that essentially defines a cohesive law. To get a priori understanding of the spatial resolution that would be required in this and similar problems, we review and combine some simple estimates of the cohesive zone sizes which correspond quite well to the sizes observed in simulations. We have assessed agreement between the methods in terms of the RMS differences in rupture time, final slip, and peak slip rate and related these to median and minimum measures of the cohesive zone resolution observed in the numerical solutions. The BI and DFM methods give virtually indistinguishable solutions to the 3-D spontaneous rupture test problem when their grid spacing Δx is small enough so that the solutions adequately resolve the cohesive zone, with at least three points for BI and at least five node points for DFM. Furthermore, grid-dependent differences in the results, for each of the two methods taken separately, decay as a power law in Δx, with the same convergence rate for each method, the calculations apparently converging to a common, grid interval invariant solution. This result provides strong evidence for the accuracy of both methods. In addition, the specific solution presented here, by virtue of being demonstrably grid-independent and consistent between two very different numerical methods, may prove useful for testing new numerical methods for spontaneous rupture problems.

Abstract: * Applications of Mickens Discretizations to Boundary Value Problems of Bratu, Gelfand and Others (R Buckmire) * Nonstandard Finite Difference Time Domain Algorithms for Computational Electromagnetics: Applications to Current Topics in Optics and Photonics (J B Cole) * Reliable Finite Difference Schemes with Applications in Mathematical Ecology (D T Dimitrov et al.) * Application of the Nonstandard Finite Difference Method in Non-Smooth Mechanics (Y Dumont) * Finite Difference Schemes on Unbounded Domains (M Ehrhardt) * Dynamically-Consistent Nonstandard Finite Difference Methods for Epidemic Models (A Gumel & K C Patidar) * Nonstandard Finite Difference Methods and Biological Models (S R-J Jang) * Contribution to the Theory of Nonstandard Finite Difference Methods and Applications to Singular Perturbation Problems (J M-S Lubuma & K C Patidar) * Nonstandard Discretization Methods on Lotka-Volterra Differential Equations (L-I W Roeger)

Abstract: The need often arises to analyze the dynamics of a system in terms of a discrete formulation. This can occur by using an a priori discrete model of the system or by discretizing a continuous model. For the latter case, the continuous model is represented by differential equations and the discrete forms come from the requirement to numerically integrate these equations. The concept of “dynamic consistency” plays an essential role in the construction of such discrete models which usually are expressed as finite difference equations. We define this concept and illustrate its application to the construction of nonstandard finite difference schemes.

Abstract: A fast algorithm is presented for solving electromagnetic scattering from a rectangular open cavity embedded in an infinite ground plane The medium inside the cavity is assumed to be (vertically) layered By introducing a transparent (artificial) boundary condition, the problem in the open cavity is reduced to a bounded domain problem A simple finite difference method is then applied to solve the model Helmholtz equation The fast algorithm is designed for solving the resulting discrete system in terms of the discrete Fourier transform in the horizontal direction, a Gaussian elimination in the vertical direction, and a preconditioning conjugate gradient method with a complex diagonal preconditioner for the indefinite interface system The existence and uniqueness of the finite difference solution are established for arbitrary wave numbers Our numerical experiments for large numbers of mesh points, up to 16 million unknowns, and for large wave numbers, eg, between 100 and 200 wavelengths, show that the algorithm is extremely efficient The cost for calculating the radar cross section, which is of significant interest in practice, is O(M2) for an $M \times M$ mesh The proposed algorithm may be extended easily to solve discrete systems from other discretization methods of the model problem

Abstract: A numerical study of the three-dimensional fluid dynamics inside a model left ventricle during diastole is presented. The ventricle is modelled as a portion of a prolate spheroid with a moving wall, whose dynamics is externally forced to agree with a simplified waveform of the entering flow. The flow equations are written in the meridian body-fitted system of coordinates, and expanded in the azimuthal direction using the Fourier representation. The harmonics of the dependent variables are normalized in such a way that they automatically satisfy the high-order regularity conditions of the solution at the singular axis of the system of coordinates. The resulting equations are solved numerically using a mixed spectral–finite differences technique. The flow dynamics is analysed by varying the governing parameters, in order to understand the main fluid phenomena in an expanding ventricle, and to obtain some insight into the physiological pattern commonly detected. The flow is characterized by a well-defined structure of vorticity that is found to be the same for all values of the parameters, until, at low values of the Strouhal number, the flow develops weak turbulence.

Abstract: Many real life problems are modelled by differential equations, for which analytical solutions are not always easy to find. One of the most difficult problems is how to solve these differential equations efficiently. Several researchers have tried to do this in various different ways (e.g. via Finite Element Methods, Standard Finite Difference Methods, Spline Approximation Methods, etc.). In recent years, to get reliable results with less effort, researchers have applied nonstandard finite difference methods (NSFDMs) and obtained competitive results to those obtained with other methods. In this survey article, the author tries to provide as much stimulating information as available regarding these NSFDMs to the researchers, which will be helpful for them as the research proceeds in this direction. While the author made the utmost efforts to include whatever he could, he would like to apologize if there are any omissions which are totally unintentional.

Abstract: A hybrid scheme composed of finite-volume and finite-difference methods is introduced for the solution of the Boussinesq equations. While the finite-volume method with a Riemann solver is applied to the conservative part of the equations, the higher-order Boussinesq terms are discretized using the finite-difference scheme. Fourth-order accuracy in space for the finite-volume solution is achieved using the MUSCL-TVD scheme. Within this, four limiters have been tested, of which van-Leer limiter is found to be the most suitable. The Adams-Basforth third-order predictor and Adams-Moulton fourth-order corrector methods are used to obtain fourth-order accuracy in time. A recently introduced surface gradient technique is employed for the treatment of the bottom slope. A new model HYWAVE, based on this hybrid solution, has been applied to a number of wave propagation examples, most of which are taken from previous studies. Examples include sinusoidal waves and bi-chromatic wave propagation in deep water, sinusoidal wave propagation in shallow water and sinusoidal wave propagation from deep to shallow water demonstrating the linear shoaling properties of the model. Finally, sinusoidal wave propagation over a bar is simulated. The results are in good agreement with the theoretical expectations and published experimental results. Copyright © 2005 John Wiley & Sons, Ltd.

Abstract: We present a parallel octree-based finite element method for large-scale earthquake ground motion simulation in realistic basins. The octree representation combines the low memory per node and good cache performance of finite difference methods with the spatial adaptivity to local seismic wavelengths characteristic of unstructured finite element methods. Several tests are provided to verify the numerical performance of the method against Green’s function solutions for homogeneous and piecewise homogeneous media, both with and without anelastic attenuation. A comparison is also provided against a finite difference code and an unstructured tetrahedral finite element code for a simulation of the 1994 Northridge Earthquake. The numerical tests all show very good agreement with analytical solutions and other codes. Finally, performance evaluation indicates excellent single-processor performance and parallel scalability over a range of 1 to 2048 processors for Northridge simulations with up to 300 million degrees of freedom. keyword: Earthquake ground motion modeling, octree, parallel computing, finite element method, elastic wave propagation

Abstract: Engineering materials are generally non-homogeneous, yet standard continuum descriptions of such materials are admissible, provided that the size of the non-homogeneities is much smaller than the characteristic length of the deformation pattern. If this is not the case, either the individual non-homogeneities have to be described explicitly or the range of applicability of the continuum concept is extended by including additional variables or degrees of freedom. In the paper the discrete nature of granular materials is modelled in the simplest possible way by means of finite-difference equations. The difference equations may be homogenised in two ways: the simplest approach is to replace the finite differences by the corresponding Taylor expansions. This leads to a Cosserat continuum theory. A more sophisticated strategy is to homogenise the equations by means of a discrete Fourier transformation. The result is a Kunin-type non-local theory. In the following these theories are analysed by considering a model consisting of independent periodic 1D chauns of solid spheres connected by shear translational and rotational springs. It is found that the Cosserat theory offers a healthy balance between accuracy and simplicity. Kunin’s integral homogenisation theory leads to a non-local Cosserat continuum description that yields an exact solution, but does not offer any real simplification in the solution of the model equations as compared to the original discrete system. The rotational degree of freedom affects the phenomenology of wave propagation considerably. When the rotation is suppressed, only one type of wave, viz. a shear wave, exists. When the restriction on particle rotation is relaxed, the velocity of this wave decreases and another, high velocity wave arises.

Abstract: This article strictly concentrates on the use of the finite-difference method (FDM) in the numerical analysis of boundary value problems associated with primarily static and to some extent quasistatic electromagnetic (EM) fields. Although all examples presented herein deal with EM-related engineering applications, some numerical aspects of FDM are also covered. The emphasis will be placed on the applications of FDM to three-dimensional boundary value problems involving objects with arbitrary geometric shapes that are composed of complex dielectric materials. Keywords: electromagnetic field; finite-difference method; boundary value; electrostatics; volumetric analysis

Abstract: A local level set algorithm for simulating interfacial flows described by the two-dimensional incompressible Navier–Stokes equations is presented. The governing equations are solved using a finite-difference discretization on a Cartesian grid and a second-order approximate projection method. The level set transport and reinitialization equations are solved in a narrow band around the interface using an adaptive refined grid, which is reconstructed every time step and refined using a simple uniform cell-splitting operation within the band. Instabilities at the border of the narrow band are avoided by smoothing the level set function in the outer part of the band. The influence of different PDE-based reinitialization strategies on the accuracy of the results is investigated. The ability of the proposed method to accurately compute interfacial flows is discussed using different tests, namely the advection of a circle of fluid in two different time-reversed vortex flows, the advection of Zalesak's rotating disk, the propagation of small-amplitude gravity and capillary waves at the interface between two superposed viscous fluids in deep water, and a classical test of Rayleigh–Taylor instability with and without surface tension effects. The interface location error and area loss for some of the results obtained are compared with those of a recent particle level set method. Copyright © 2005 John Wiley & Sons, Ltd.

Abstract: A numerical study of mixed convection in a vertical channel filled with a porous medium including the effect of inertial forces is studied by taking into account the effect of viscous and Darcy dissipations. The flow is modeled using the Brinkman- Forchheimer-extended Darcy equations. The two boundaries are considered as isothermal- isothermal, isoflux-isothermal and isothermal-isoflux for the left and right walls of the channel and kept either at equal or at different temperatures. The governing equations are solved numerically by finite difference method with Southwell-Over-Relaxation technique for extended Darcy model and analytically using perturbation series method for Darcian model. The velocity and temperature fields are obtained for various porous parameter, inertia effect, product of Brinkman number and Grashof number and the ratio of Gras- hof number and Reynolds number for equal and different wall temperatures. Nusselt num- ber at the walls is also determined for three types of thermal boundary conditions. The viscous dissipation enhances the flow reversal in the case of downward flow while it coun- ters the flow in the case of upward flow. The Darcy and inertial drag terms suppress the flow. It is found that analytical and numerical solutions agree very well for the Darcian model.

Abstract: In this paper, we discuss the parameter-uniform finite difference method for a coupled system of singularly perturbed convection–diffusion equations. The leading term of each equation is multiplied by a small but different magnitude positive parameter, which leads to the overlap and interact boundary layer. We analyze the boundary layer and construct a piecewise-uniform mesh on the variant of the Shishkin mesh. We prove that our schemes converge almost first-order uniformly with respect to small parameters. We present some numerical experiments to support our theoretical analysis.