Abstract: Solving efficiently the wave equations involved in modeling acoustic, elastic or electromagnetic wave propagation remains a challenge both for research and industry. To attack the problems coming from the propagative character of the solution, the author constructs higher-order numerical methods to reduce the size of the meshes, and consequently the time and space stepping, dramatically improving storage and computing times. This book surveys higher-order finite difference methods and develops various mass-lumped finite (also called spectral) element methods for the transient wave equations, and presents the most efficient methods, respecting both accuracy and stability for each sort of problem. A central role is played by the notion of the dispersion relation for analyzing the methods. The last chapter is devoted to unbounded domains which are modeled using perfectly matched layer (PML) techniques. Numerical examples are given.

Abstract: In recent years high order numerical methods have been widely used in computational fluid dynamics (CFD), to effectively resolve complex flow features using meshes which are reasonable for today''s computers. In this paper we review and compare three types of high order methods being used in CFD, namely the weighted essentially non-oscillatory (WENO) finite difference methods, the WENO finite volume methods, and the discontinuous Galerkin (DG) finite element methods. We summarize the main features of these methods, from a practical user''s point of view, indicate their applicability and relative strength, and show a few selected numerical examples to demonstrate their performance on illustrative model CFD problems.

Abstract: We propose a stable and conservative finite difference scheme to solve numerically the Cahn-Hilliard equation which describes a phase separation phenomenon. Numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly ill-posed problem. We design a new difference scheme based on a general strategy proposed recently by Furihata and Mori. The new scheme inherits characteristic properties, the conservation of mass and the decrease of the total energy, from the equation. The decrease of the total energy implies boundedness of discretized Sobolev norm of the solution. This in turn implies, by discretized Sobolev's lemma, boundedness of max norm of the solution, and hence the stability of the solution. An error estimate for the solution is obtained and the order is \(O( (\Delta x)^2 + (\Delta t)^2)\). Numerical examples demonstrate the effectiveness of the proposed scheme.

Abstract: We formalize the transfer of essential properties of the solution of a differential equation to the solution of a discrete scheme as qualitative stability with respect to the properties. This permits us to motivate some rules (viz. on the order of the difference equation, on the renormalization of the denominator of the discrete derivative, and on nonlocal approximation of nonlinear terms) used in the design of nonstandard finite difference schemes. Extensions of some models are considered, and numerical examples confirming the efficiency of the nonstandard approach are provided. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 518–543, 2001

Abstract: New finite difference methods using Cartesian grids are developed for elliptic interface problems with variable discontinuous coefficients, singular sources, and nonsmooth or even discontinuous solutions. The new finite difference schemes are constructed to satisfy the sign property of the discrete maximum principle using quadratic optimization techniques. The methods are shown to converge under certain conditions using comparison functions. The coefficient matrix of the resulting linear system of equations is an M-matrix and is coupled with a multigrid solver. Numerical examples are also provided to show the efficiency of the proposed methods.

Abstract: In this paper, we present a simple yet accurate conformal Finite Difference Time Domain (FDTD) technique, which can be used to analyze curved dielectric surfaces. Unlike the existing conformal techniques for handling dielectrics, the present approach utilizes the individual electric field component along the edges of the cell, rather than requiring the calculation of its area or volume, which is partially filled with a dielectric material. The new technique shows good agreement with the results derived by Mode Matching and analytical methods.

Abstract: Based on a regularized Volterra equation, two different approaches for numeri- cal differentiation are considered. The first approach consists of solving a regularized Volterra equation while the second approach is based on solving a disretized version of the regularized Volterra equation. Numerical experiments show that these methods are efficient and compete fa- vorably with the variational regularization method for stable calculating the derivatives of noisy functions.

Abstract: The problem of coupled heat and mass transfer by natural convection from a semi-infinite inclined flat plate in the presence of an external magnetic field and internal heat generation or absorption effects is formulated. The plate surface has a power-law variation of both wall temperature and concentration and is permeable to allow for possible fluid wall suction or blowing. The resulting governing equations are transformed using a similarity transformation and then solved numerically by an implicit, iterative, finite-difference scheme. Comparisons with previously published work are performed and good agreement is obtained. A parametric study of all involved parameters is conducted and a representative set of numerical results for the velocity and temperature profiles as well as the skin-friction parameter, average Nusselt number, and the average Sherwood number is illustrated graphically to show typical trends of the solutions.

Abstract: The development of velocity and temperature fields of an incompressible viscous electrically conducting fluid, caused by an impulsive stretching of the surface in two lateral directions and by suddenly increasing the surface temperature from that of the surrounding fluid, is studied. The partial differential equations governing the unsteady laminar boundary-layer flow are solved numerically using an implicit finite difference scheme. For some particular cases, closed form solutions are obtained, and for large values of the independent variable asymptotic solutions are found. The surface shear stresses inx-andy-directions and the surface heat transfer increase with the magnetic field and the stretching ratio, and there is a smooth transition from the short-time solution to the long-time solution.

Abstract: This paper addresses the problem of stability analysis of finite-difference time-domain (FDTD) approximations for Maxwell's equations. The combination of the von Neumann method with the Routh-Hurwitz criterion is proposed as an algebraic procedure for obtaining analytical closed-form stability expressions. This technique is applied to the problem of determining the stability conditions of an extension of the FDTD method to incorporate dispersive media previously reported in the literature. Both Debye and Lorentz dispersive media are considered. It is shown that, for the former case, the stability limit of the conventional FDTD method is preserved. However, for the latter case, a more restrictive stability limit is obtained. To overcome this drawback, a new scheme is presented, which allows the stability limit of the conventional FDTD method to be maintained.

Abstract: A high-order method is used to perform large-eddy simulations of a supersonic compression-ramp flowfield. The procedure employs an implicit approximately factored finite difference algorithm, which is used in conjunction with a 10th-order nondispersive filter. Spatial derivatives are approximated by a sixth-order compact scheme, and Newton-like subiterations are applied to achieve second-order temporal accuracy. In the region of strong shock waves, the compact differencing of convective fluxes is replaced locally by an upwind-biased scheme. Both the Smagorinsky and dynamic subgrid-scale stress models are incorporated in the simulations. Details of the method are summarized, and a number of computations are carried out. Comparisons are made between the respective solutions as well as with available experimental data and with previous numerical results

Abstract: A new explicit fourth-order finite-difference time-domain (FDTD) scheme for three-dimensional electromagnetic field simulation is proposed in this paper. A symplectic integrator propagator, which is also known as a decomposition of the exponential operator or a general propagation technique, is directly applied to Maxwell's equations in the scheme. The scheme is nondissipative and saves memory. The Courant stability limit of the scheme is 30% larger than that of the standard FDTD method. The perfectly matched layer absorbing boundary condition is applicable to the scheme. A specific eigenmode of a waveguide is successfully excited in the scheme. Stable and accurate performance is demonstrated by numerical examples.

Abstract: A particle velocity-strain, finite-difference (FD) method with a perfectly matched layer (PML) absorbing boundary condition is developed for the simulation of elastic wave propagation in multidimensional heterogeneous poroelastic media. Instead of the widely used second-order differential equations, a first-order hyperbolic leap-frog system is obtained from Biot’s equations. To achieve a high accuracy, the first-order hyperbolic system is discretized on a staggered grid both in time and space. The perfectly matched layer is used at the computational edge to absorb the outgoing waves. The performance of the PML is investigated by calculating the reflection from the boundary. The numerical method is validated by analytical solutions. This FD algorithm is used to study the interaction of elastic waves with a buried land mine. Three cases are simulated for a mine-like object buried in “sand,” in purely dry “sand” and in “mud.” The results show that the wave responses are significantly different in these cases. The target can be detected by using acoustic measurements after processing.

Abstract: We study the $L^1$ -stability and error estimates of general approximate solutions for the Cauchy problem associated with multidimensional Hamilton-Jacobi (H-J) equations. For strictly convex Hamiltonians, we obtain a priori error estimates in terms of the truncation errors and the initial perturbation errors. We then demonstrate this general theory for two types of approximations: approximate solutions constructed by the vanishing viscosity method, and by Godunov-type finite difference methods. If we let $\epsilon$ denote the `small scale' of such approximations (– the viscosity amplitude $\epsilon$ , the spatial grad-size $\Delta x$ , etc.), then our $L^1$ -error estimates are of ${\cal O}(\epsilon)$ , and are sharper than the classical $L^\infty$ -results of order one half, ${\cal O}(\sqrt{\epsilon})$ . The main building blocks of our theory are the notions of the semi-concave stability condition and $L^1$ -measure of the truncation error. The whole theory could be viewed as a multidimensional extension of the $Lip^\prime$ -stability theory for one-dimensional nonlinear conservation laws developed by Tadmor et. al. [34,24,25]. In addition, we construct new Godunov-type schemes for H-J equations which consist of an exact evolution operator and a global projection operator. Here, we restrict our attention to linear projection operators (first-order schemes). We note, however, that our convergence theory applies equally well to nonlinear projections used in the context of modern high-resolution conservation laws. We prove semi-concave stability and obtain $L^1$ -bounds on their associated truncation errors; $L^1$ -convergence of order one then follows. Second-order (central) Godunov-type schemes are also constructed. Numerical experiments are performed; errors and orders are calculated to confirm our $L^1$ -theory.

Abstract: Linear finite difference inequalities nonlinear finite difference inequalities nonlinear finite difference inequalities II linear multidimensional finite difference inequalities nonlinear multidimensional finite difference inequalities.

Abstract: A three-dimensional numerical model based on the complete Navier-Stokes equations is developed and presented in this paper. The model can be used for the problem of propagation of fully nonlinear water waves. The Navier-Stokes equations are first transformed from an irregular calculation domain to a regular one using sigma coordinates. The projection method is used to separate advection and diffusion terms from the pressure terms in Navier-Stokes equations. MacCormack's explicit scheme is used for the advection and diffusion terms, and it has second-order accuracy in both space and time. The pressure variable is further separated into hydrostatic and hydrodynamic pressures so that the computer rounding errors can be largely avoided. The resulting hydrodynamic pressure equation is solved by a multigrid method. A staggered mesh and central spatial finite-difference scheme are used. The model is tested against the experimental data of Luth et al., and the comparison shows that higher harmonics can be modeled well. Comparison of the model solutions with the elliptic shoal case confirms that the present model works well for wave refraction and diffraction with strong wave focusing.

Abstract: A numerical method for solving three-dimensional free surface flows is presented. The technique is an extension of the GENSMAC code for calculating free surface flows in two dimensions. As in GENSMAC, the full Navier-Stokes equations are solved by a finite difference method; the fluid surface is represented by a piecewise linear surface composed of quadrilaterals and triangles containing marker particles on their vertices; the stress conditions on the free surface are accurately imposed; the conjugate gradient method is employed for solving the discrete Poisson equation arising from a velocity update; and an automatic time step routine is used for calculating the time step at every cycle. A program implementing these features has been interfaced with a solid modelling routine defining the flow domain. A user-friendly input data file is employed to allow almost any arbitrary three-dimensional shape to be described. The visualization of the results is performed using computer graphic structures such as phong shade, flat and parallel surfaces. Results demonstrating the applicability of this new technique for solving complex free surface flows, such as cavity filling and jet buckling, are presented.

Abstract: In the article, a finite difference scheme for the numerical integration of the Landau–Lifshitz equation is presented. The scheme is based on the application of the midpoint rule coupled with a suitable extrapolation formula. The important properties of the scheme are the conservation of magnetization magnitude, its linearity, its second order truncation error accuracy, and the absence of spatial coupling. The accuracy of the scheme has been extensively tested by comparing numerical solutions with exact analytical solutions and by applying the scheme to the analysis of magnetization dynamics in conducting thin films.

Abstract: We develop here a finite difference approach for valuing a discretely sampled variance swap within a Black-Scholes framework. This approach incorporates the observed volatility skew and is capable of handling various definitions of the variance. It is benchmarked against Monte-Carlo simulation in the presence of a volatility skew and is shown to provide extremely accurate values for a variance swap. Our method is based on decomposing the problem of valuing a variance swap into a set of one-dimensional PDE problems, each of which is then solved using a finite difference method.