Abstract: SUMMARY A new algorithm for volume tracking which is based on the concept of flux-corrected transport (FCT) is introduced. It is applicable to incompressible 2D flow simulations on finite volume and difference meshes. The method requires no explicit interface reconstruction, is direction-split and can be extended to 3D and orthogonal curvilinear meshes in a straightforward manner. A comparison of the new scheme against well-known existing 2D finite volume techniques is undertaken. A series of progressively more difficult advection tests is used to test the accuracy of each scheme and it is seen that simple advection tests are inadequate indicators of the performance of volume-tracking methods. A straightforward methodology is presented that allows more rigorous estimates to be made of the error in volume advection and coupled volume and momentum advection in real flow situations. The volume advection schemes are put to a final test in the case of Rayleigh‐Taylor instability. 1997 by CSIRO. In the numerical computation of multifluid problems such as density currents or Rayleigh‐Taylor instability there is a need for an accurate representation of the interface separating two immiscible fluids. Free surface flows such as water waves and splashing droplets are an approximation to the multifluid problem in which one of the fluids (usually a gas) is neglected as having an insignificant influence on the dynamics of the system. In a general free surface flow problem, fluid coalescence and detachment may occur and deforming meshes cannot be used. In this case the need of an accurate and sharp interface is even greater than in true multifluid computations. Although a slightly diffuse interface may be acceptable in a problem where the continuity, momentum and energy equations are solved throughout the entire mesh, in a free surface simulation the location of the interface determines the size and shape of the computational domain and specifies where boundary conditions must be applied. In this case a diffuse interface cannot be tolerated. On finite volume (or difference) meshes, standard advection techniques can be used in multifluid problems to advect either the density or a material indicator function, however these methods are either diffusive (e.g. first order upwinding) or unstable (higher order schemes in which unphysical oscillations appear in the vicinity of the interface). Numerous techniques have been devised to limit the diffusiveness of low order schemes and to minimize the instability of high order schemes (see e.g.

Abstract: A second-order accurate interface tracking method for the solution of incompressible Stokes flow problems with moving interfaces on a uniform Cartesian grid is presented. The interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids. The interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed. The Stokes equations are then discretized using the second-order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper [SIAM J. Numer. Anal., 31(1994), pp. 1019--1044]. The resulting velocities are interpolated to the interface to determine the motion of the interface. An implicit quasi-Newton method is developed that allows reasonable time steps to be used.

Abstract: We present an expanded mixed finite element approximation of second-order elliptic problems containing a tensor coefficient. The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, the negative of its gradient, and its flux (the tensor coefficient times the negative gradient). The resulting linear system is a saddle point problem. In the case of the lowest order Raviart--Thomas elements on rectangular parallelepipeds, we approximate this expanded mixed method by incorporating certain quadrature rules. This enables us to write the system as a simple, cell-centered finite difference method requiring the solution of a sparse, positive semidefinite linear system for the scalar unknown. For a general tensor coefficient, the sparsity pattern for the scalar unknown is a 9-point stencil in two dimensions and 19 points in three dimensions. Existing theory shows that the expanded mixed method gives optimal order approximations in the $L^2$- and $H^{-s}$-norms (and superconvergence is obtained between the $L^2$-projection of the scalar variable and its approximation). We show that these rates of convergence are retained for the finite difference method. If $h$ denotes the maximal mesh spacing, then the optimal rate is $O(h)$. The superconvergence rate $O(h^{2})$ is obtained for the scalar unknown and rate $O(h^{3/2})$ for its gradient and flux in certain discrete norms; moreover, the full $O(h^{2})$ is obtained in the strict interior of the domain. Computational results illustrate these theoretical results.

Abstract: A novel conformal finite-difference time-domain (CFDTD) technique for locally distorted contours that accurately model curved metallic objects is presented in this paper. This approach is easy to implement and is numerically stable. Several examples are presented to demonstrate that the new method yields results that are far more accurate than those generated by the conventional staircasing approach. Example geometries include cylindrical and spherical cavities, and a circular microstrip patch antenna. Accuracy of the scheme is demonstrated by comparing the results derived from analytical and Method of Moments (MoM) techniques.

Abstract: We have simulated flow past a circular cylinder at a Reynolds number of 3.9 X 10 3 using a solver that employs an energy-conservative second-order central difference scheme for spatial discretization. Detailed comparisons of turbulence statistics and energy spectra in the downstream wake region (7.0 < x/D < 10.0) have been made with the results of Beaudan and Moin and with experiments to assess the impact of numerical diffusion on the flowfield. Based on these comparisons, conclusions are drawn on the suitability of higher-order upwind schemes for LES in complex geometries.

Abstract: Picard-Vessiot rings.- Algorithms for difference equations.- The inverse problem for difference equations.- The ring S of sequences.- An excursion in positive characteristic.- Difference modules over .- Classification and canonical forms.- Semi-regular difference equations.- Mild difference equations.- Examples of equations and galois groups.- Wild difference equations.- q-difference equations.

Abstract: A flexible and simple way of introducing stress-free boundary conditions for including three-dimensional (3D) topography in the finite-difference method is presented. The 3D topography is discretized in a staircase by stacking unit material cells in a staggered-grid scheme. The shear stresses are distributed on the 12 edges of the unit material cell so that only shear stresses appear on the free surface and normal stresses always remain embedded within the solid region. This configuration makes it possible to implement stress-free boundary conditions at the free surface by setting the Lame coefficients λ and μ to zero without generating any physically unjustified condition. Arbitrary 3D topographies are realized by changing the distribution of λ and μ in the computational domain. Our method uses a parsimonious staggered-grid scheme that requires only 3/4 of the memory used in the conventional staggered-grid scheme in which six stress components and three velocity components need to be stored. Numerical tests indicate that 25 grids per wavelength are required for stable calculation. The finite-difference results are compared with those of the boundary-element method for the two-dimensional (2D) semi-circular canyon model. We also present the responses of a segment of semi-circular canyon and hemispherical cavity to vertically incident plane P, SV , and SH waves and discuss the response of a Gaussian hill to an isotropic point source embedded in the hill. In the segment of semi-circular canyon, the later portions of the synthetics are characterized by phases scattered from the two vertical side walls. The hemispherical cavity and 2D semi-circular canyon both show focusing of energy at the bottom of the cavity, although the focusing effect is stronger in the former geometry. Focusing and defocusing effects due to the strong topography of the Gaussian hill produce a strong amplification of displacements at a spot located on the flank opposite to the source. Backscattering from the top of the hill is also clearly seen.

Abstract: A general formulation is presented for finite-difference time-domain (FDTD) modeling of wave propagation in arbitrary frequency-dispersive media. Two algorithmic approaches are outlined for incorporating dispersion into the FDTD time-stepping equations. The first employs a frequency-dependent complex permittivity (denoted Form-1), and the second employs a frequency-dependent complex conductivity (denoted Form-2). A Pade representation is used in Z-transform space to represent the frequency-dependent permittivity (Form-1) or conductivity (Form-2). This is a generalization over several previous methods employing either Debye, Lorentz, or Drude models. The coefficients of the Pade model may be obtained through an optimization process, leading directly to a finite-difference representation of the dispersion relation, without introducing discretization error. Stability criteria for the dispersive FDTD algorithms are given. We show that several previously developed dispersive FDTD algorithms can be cast as special cases of our more general framework. Simulation results are presented for a one-dimensional (1-D) air/muscle example considered previously in the literature and a three-dimensional (3-D) radiation problem in dispersive, lossy soil using measured soil data.

Abstract: Volume-averaged equations are developed governing steady, laminar, fully developed, hydromagnetic mixed convection non-Darcian flow of an electrically conducting and heat-generating / absorbing fluid in a channel embedded in a uniform porous medium. Proper dimensionless parameters are employed for various thermal boundary conditions on the left and right walk of the channel prescribed as isothermal-isothermal, isothermal-iso-flux, and isoflux-isothermal. Analytical expressions for the velocity and temperature profiles in the channel as well as for the mass flow rate, friction factor, and heat carried out by the fluid in the channel are developed for special cases of the problem. Conditions for the occurrence of fluid backflow zones are reported. The fully nonlinear governing equations are solved numerically by an implicit finite difference method. Favorable comparisons with the developed analytical results and previously published work are performed. Graphical results of the closed-form and numer...

Abstract: A method is presented for adaptively solving hyperbolic PDEs. The method is based on an interpolating wavelet transform using polynomial interpolation on dyadic grids. The adaptability is performed automatically by thresholding the wavelet coefficients. Operations such as differentiation and multiplication are fast and simple due to the one-to-one correspondence between point values and wavelet coefficients in the interpolating basis. Treatment of boundary conditions is simplified in this sparse point representation (SPR). Numerical examples are presented for one- and two-dimensional problems. It is found that the proposed method outperforms a finite difference method on a uniform grid for certain problems in terms of flops.

Abstract: SUMMARY An algorithm for the numerical modelling of magnetotelluric fields in 2-D generally anisotropic block structures is presented. Electrical properties of the individual homogeneous blocks are described by an arbitrary symmetric and positive-definite conductivity tensor. The problem leads to a coupled system of partial differential equations for the strike-parallel components of the electromagnetic field, Ex and H,. These equations are numerically approximated by the finite-difference (FD) method, making use of the integro-interpolation approach. As the magnetic component H, is constant in the non-conductive air, only equations for the electric mode are approximated within the air layer. The system of linear difference equations, resulting from the FD approximation, can be arranged in such a way that its matrix is symmetric and band-limited, and can be solved, for not too large models, by Gaussian elimination. The algorithm is applied to model situations which demonstrate some non-trivial phenomena caused by electrical anisotropy. In particular, the effect of 2-D anisotropy on the relation between magnetotelluric impedances and induction arrows is studied in detail.

Abstract: A compact central-difference approximation in conjunction with the Yee (1966) grid is used to compute the spatial derivatives in Maxwell's equations. To advance the semi-discrete equations, the four-stage Runge-Kutta (RK) integrator is invoked. This combination of spatial and temporal differencing leads to a scheme that is fourth-order accurate, conditionally stable, and highly efficient. Moreover, the use of compact differencing allows one to apply the compact operator in the vicinity of a perfect conductor-an attribute not found in other higher order methods. Results are provided that quantify the spectral properties of the method. Simulations are conducted on problem spaces that span one and three dimensions and whose domains are of the open and closed type. Results from these simulations are compared with exact closed-form solutions; the agreement between these results is consistent with numerical analysis.

Abstract: Grain drying is a simultaneous heat and moisture transfer problem. The modelling of such a problem is of significance in understanding and controlling the drying process. In the present study, a mathematical model for coupled heat and moisture transfer problem is presented. The model consists of four partial differential equations for mass balance, heat balance, heat transfer and drying rate. A simple finite difference method is used to solve the equations. The method shows good flexibility in choosing time and space steps which enable the simulation of long term grain drying/cooling processes. A deep barley bed is used as an example of grain beds in the current simulation. The results are verified against experimental data taken from literature. The analysis of the effects of operating conditions on the temperature and moisture content within the bed is also carried out

Abstract: Natural convection flow of an absorbing fluid up a uniform porous medium supported by a semi-infinite, ideally transparent, vertical flat plate due to solar radiation is considered. Boundary-layer equations are derived using the usual Boussinesq approximation and accounting for applied incident radiation flux. A convection type boundary condition is used at the plate surface. These equations exhibit no similarity solution. However, the local similarity method is employed for the solution of the present problem so as to allow comparisons with previously published work. The resulting approximate nonlinear ordinary differential equations are solved numerically by a standard implicit iterative finite-difference method. Graphical results for the velocity and temperature fields as well as the boundary friction and Nusselt number are presented and discussed.

Abstract: The technique of Richardson extrapolation, which has previously been used on time-independent problems, is extended so that it can also be used on time-dependent problems. The technique presented is completed in the sense that the extrapolated solution is calculated at all spatial grid nodes which coincide with nodes of the finest grid considered. Numerical examples are presented when the technique is applied to the Lax–Wendroff and Crank–Nicholson finite difference schemes which are used to approximate solutions to the convection–diffusion equation. The examples show that extrapolation can be an easy and efficient way in which to produce accurate numerical solutions to time-dependent problems. © 1997 John Wiley & Sons, Ltd.

Abstract: Two moving mesh partial differential equations (MMPDEs) with spatial smoothing are derived based upon the equidistribution principle. This smoothing technique is motivated by the robust moving mesh method of Dorfi and Drury [J. Comput. Phys., 69 (1987), pp. 175--195]. It is shown that under weak conditions the basic property of no node-crossing is preserved by the spatial smoothing, and a local quasi-uniformity property of the coordinate transformations determined by these MMPDEs is proven. It is also shown that, discretizing the MMPDEs using centered finite differences, these basic properties are preserved.

Abstract: A hybrid finite-difference time-domain (FDTD) method is proposed for solving transient electromagnetic problems associated with structures of curved surfaces. The method employs the conventional FDTD method for most of the regular region but introduces the tetrahedral edge-based finite-element scheme to model the region near the curved surfaces. Without any interpolation for the fields on the curved surface, nor any additional stability constraint due to the finer division near the curved surfaces, the novel finite-element scheme is found to have second-order accuracy, unconditional stability, programming ease, and computational efficiency. The hybrid method is applied to solve the electromagnetic scattering of three-dimensional (3-D) arbitrarily shaped dielectric objects to demonstrate its superior performance.

Abstract: Two diffusional models were developed for finite cylindrical shaped bodies which take into account that mass transfer could have a nonisotropic nature. In the first model, sample shrinkage was ignored; thus, it was solved by the separation of variables method. This hypothesis of constant sample volume was not assumed in the second model, which solved mass transfer equations through a finite difference scheme. The proposed models were applied to the simulation of drying curves of green beans (Phaseolus vulgaris). Two different effective diffusivity coefficients, one radial and the other axial, as a consequence of the mass transfer through both directions, were estimated in each model. The effective diffusivities estimated with the proposed models varied with the temperature according to the Arrhenius law. The average percentage of variance explained by the fixed boundaries model was 96.1% and increased to 99.1% when shrinkage was considered (in the model solved by a finite difference method). Keywords: Dry...

Abstract: The development and application of a one-dimensional heat transport model in highly transient streams governed by unsteady flow is described here. The resultant framework consists of two modules called the hydrodynamic and heat transport modules. In the hydrodynamic module, the hydraulic variables such as flow depth and velocity are simulated. Based on this information, the heat transport module is executed to calculate temperatures. A new approach—coupling heat transport in the surface water and diffusion in the sediment zone—is developed and applied for this module. In this approach, an interaction term accounts for the flux of heat energy between the water and sediments, which affects the distribution of water temperature in clear and shallow streams. Implicit finite difference methods called the Preissmann four-point and the Crank-Nicolson schemes are used to solve each module. Application of this framework demonstrates that the model effectively simulates the hydraulic variables and temperatures.

Abstract: The present work concerns the numerical treatment of fretting in the interface between a body and a rigid foundation. Starting from a variational formulation of a fretting model given in a framework of continuum thermodynamics, an augmented Lagrangian formulation is derived by introducing finite element discretizations in space and a finite difference discretization in time. The augmented Lagrangian formulation is implemented and solved by a Newton method for the two-dimensional case.

Abstract: A free convertion flow of an optically dense viscous incompressible fluid along a vertical thin circular cylinder has been studied with effect of radiation when the surface temperature is uniform. With appropriate transformations, the boundary layer equations governing the flow are reduced to local nonsimilarity equations. Solutions of the governing equations are obtained employing the implicit finite difference methods together with Keller box scheme as well the local nonsimilarity method with second order truncation for all ξ (nondimensional transverse curvature parameter) in the interval [0,10] and are expressed in terms of local Nusselt number for a range of values of the pertinent parameters. Effects of pertinent parameters, such as, the radiation parameter, R d , the surface temperature parameter, θ w , taking Prandtl number, Pr, equals 0.7 on the velocity and temperature field are also presented graphically. From the solution it is seen that increase of R d , or θ w leads to increase in the local rate of heat transfer coefficients. Results obtained by both the methods are obtained in excellent agreement between each other upto ξ = 10.

Abstract: SUMMARY We present new finite difference schemes for the incompressible Navier‐Stokes equations. The schemes are based on two spatial differencing methods; one is fourth-order-accurate and the other is sixth-order accurate. The temporal differencing is based on backward differencing formulae. The schemes use non-staggered grids and satisfy regularity estimates, guaranteeing smoothness of the solutions. The schemes are computationally efficient. Computational results demonstrating the accuracy are presented. 1997 by John Wiley & Sons, Ltd. High-order-accurate finite difference schemes are important in scientific computation because they offer a means to obtain accurate solutions with less work than may be required for methods of lower accuracy. Finite difference methods are attractive because of the relative ease of implementation and flexibility. In this paper we present new finite difference schemes for the incompressible Navier‐Stokes equations. The schemes are based on two spatial differencing methods, one a fourth-order-accurate method and one a sixth-order-accurate method. There are several temporal differencing methods presented in Section 7. These temporal schemes can be used with either of the spatial differencing methods. The temporal differencing is based on backward differencing formula (BDF) schemes that are used for stiff ordinary differential equations. The schemes are implicit and appear to be unconditionally stable for the Stokes equations. (A rigorous stability analysis is the subject of further research.) High-order methods have been presented by Rai and Moin 1 and Lele 2 for the fractional step method proposed by Kim and Moin, 3 There is an excellent study of these methods in the paper by Tafti. 4 A disadvantage of these methods is that because they are explicit, there is a severe stability limit on the time step. Moreover, as pointed out by Perot, 5 the pressure for fractional step methods can be no better than first-order-accurate in time. Projection methods also have difficulty with higherorder accuracy in time. 6 This is not so for the methods presented here, where the pressure can be determined to a high order of accuracy. For steady flows the method of Aubert and Deville 7 can be applied to yield fourth-order accuracy, at the expense of increasing the number of unknowns and computational complexity of the system. All these methods use staggered meshes. The schemes presented in this paper are for orthogonal Cartesian grids on non-staggered grids, the velocity components and pressure unknowns are assigned to a common grid. The schemes are for the