Showing papers on "Finite difference published in 1990"
TL;DR: In this paper, the authors used MQ as the spatial approximation scheme for parabolic, hyperbolic and the elliptic Poisson's equation, and showed that MQ is not only exceptionally accurate, but is more efficient than finite difference schemes which require many more operations to achieve the same degree of accuracy.
Abstract: This paper is the second in a series of investigations into the benefits of multiquadrics (MQ). MQ is a true scattered data, multidimensional spatial approximation scheme. In the previous paper, we saw that MQ was an extremely accurate approximation scheme for interpolation and partial derivative estimates for a variety of two-dimensional functions over both gridded and scattered data. The theory of Madych and Nelson shows for the space of all conditionally positive definite functions to which MQ belongs, a semi-norm exists which is minimized by such functions. In this paper, MQ is used as the spatial approximation scheme for parabolic, hyperbolic and the elliptic Poisson's equation. We show that MQ is not only exceptionally accurate, but is more efficient than finite difference schemes which require many more operations to achieve the same degree of accuracy.
1,914 citations
01 Jan 1990
TL;DR: In this article, the authors presented an enhanced multiquadrics (MQ) scheme for spatial approximations, which is a true scattered data, grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions.
Abstract: A~traet--We present a powerful, enhanced multiquadrics (MQ) scheme developed for spatial approximations. MQ is a true scattered data, grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions. It is continuously differentiable and integrable and is capable of representing functions with steep gradients with very high accuracy. Monotonicity and convexity are observed properties as a result of such high accuracy. Numerical results show that our modified MQ scheme is an excellent method not only for very accurate interpolation, but also for partial derivative estimates. MQ is applied to a higher order arbitrary Lagrangian-Eulerian (ALE) rezoning. In the second paper of this series, MQ is applied to parabolic, hyperbolic and elliptic partial differential equations. The parabolic problem uses an implicit time-marching scheme whereas the hyperbolic problem uses an explicit time-marching scheme. We show that MQ is also exceptionally accurate and efficient. The theory of Madych and Nelson shows that the MQ interpolant belongs to a space of functions which minimizes a semi-norm and gives credence to our results. 1. BACKGROUND The study of arbitrarily shaped curves, surfaces and bodies having arbitrary data orderings has immediate application to computational fluid-dynamics. The governing equations not only include source terms but gradients, divergences and Laplacians. In addition, many physical processes occur over a wide range of length scales. To obtain quantitatively accurate approximations of the physics, quantitatively accurate estimates of the spatial variations of such variables are required. In two and three dimensions, the range of such quantitatively accurate problems possible on current multiprocessing super computers using standard finite difference or finite element codes is limited. The question is whether there exist alternative techniques or combinations of techniques which can broaden the scope of problems to be solved by permitting steep gradients to be modelled using fewer data points. Toward that goal, our study consists of two parts. The first part will investigate a new numerical technique of curve, surface and body approximations of exceptional accuracy over an arbitrary data arrangement. The second part of this study will use such techniques to improve parabolic, hyperbolic or elliptic partial differential equations. We will demonstrate that the study of function approximations has a definite advantage to computational methods for partial differential equations. One very important use of computers is the simulation of multidimensional spatial processes. In this paper, we assumed that some finite physical quantity, F, is piecewise continuous in some finite domain. In many applications, F is known only at a finite number of locations, {xk: k = 1, 2 ..... N} where xk = x~ for a univariate problem, and Xk = (x~,yk .... )X for the multivariate problem. From a finite amount of information regarding F, we seek the best approximation which can not only supply accurate estimates of F at arbitrary locations on the domain, but will also provide accurate estimates of the partial derivatives and definite integrals of F anywhere on the domain. The domain of F will consist of points, {xk }, of arbitrary ordering and sub-clustering. A rectangular grid is a very special case of a data ordering. Let us assume that an interpolation function, f, approximates F in the sense that
1,764 citations
TL;DR: In this article, a class of second-order conservative finite difference algorithms for solving numerically time-dependent problems for hyperbolic conservation laws in several space variables is presented, in which the numerical fluxes are obtained by solving the characteristic form of the full multidimensional equations at the zone edge, and all fluxes were evaluated and differenced at the same time.
829 citations
TL;DR: In this article, the use of high speed, high capacity vector computers allows the resultant finite-difference equations to be factored in-place, allowing inversions to be generated using data from a very large number of source positions.
Abstract: Frequency-domain methods are well suited to the imaging of wide-aperture cross-hole data. However, although the combination of the frequency domain with the wavenumber domain has facilitated the development of rapid algorithms, such as diffraction tomography, this has also required linearization with respect to homogeneous reference media. This restriction, and association restrictions on source-receiver geometries, are overcome by applying inverse techniques that operate in the frequency-space domain.
In order to incorporate the rigorous modelling technique of finite differences into the inverse procedure a nonlinear approach is used. To reduce computational costs the method of finite differences is applied directly to the frequency-domain wave equation. The use of high speed, high capacity vector computers allow the resultant finite-difference equations to be factored in-place. In this way wavefields can be computed for additional source positions at minimal extra cost, allowing inversions to be generated using data from a very large number of source positions.
Synthetic studies show that where weak scatter approximations are valid, diffraction tomography performs slightly better than a single iteration of non-linear inversion. However, if the background velocities increase systematically with depth, diffraction tomography is ineffective whereas non-linear inversion yields useful images from one frequency component of the data after a single iteration. Further synthetic studies indicate the efficacy of the method in the time-lapse monitoring of injection fluids in tertiary hydrocarbon recovery projects.
567 citations
TL;DR: In this article, a finite-difference beam propagation method (FD-BPM) is outlined and assessed in comparison with conventional beam propagation methods which use fast Fourier transformation.
Abstract: A finite-difference beam propagation method (FD-BPM) is outlined and assessed in comparison with a conventional beam propagation method (FFT-BPM) which uses fast Fourier transformation. In the comparative study three straight waveguides with different index profiles that are frequently encountered in integrated optics are utilized. Using both methods normalized effective index values of the eigenmodes of these waveguides are calculated and compared with the exact values obtained from analytical expressions. As a further accuracy criterion, the power loss due to numerical errors when an eigenmode of a waveguide is excited is evaluated. Based on this comparison the accuracy, computational efficiency, and stability of the FD-BPM are assessed. >
508 citations
TL;DR: The approach can be used to value any derivative security dependent on a single state variable and can be extended to deal with many derivative security pricing problems where there are several state variables.
Abstract: This paper suggests a modification to the explicit finite difference method for valuing derivative securities. The modification ensures that, as smaller time intervals are considered, the calculated values of the derivative security converge to the solution of the underlying differential equation. It can be used to value any derivative security dependent on a single state variable and can be extended to deal with many derivative security pricing problems where there are several state variables. The paper illustrates the approach by using it to value bonds and bond options under two different interest rate processes.
377 citations
TL;DR: In this article, a system of differential equations that describe the growth of a forest fire front in time for variable fuel, weather and topographical conditions is derived, and a finite difference solution is presented, together with its inherent problems and their solution.
Abstract: A system of differential equations that describe the growth of a forest fire front in time for variable fuel, weather and topographical conditions is derived. The system is first order, non-linear and contains parameters that can be obtained from forestry data. A finite difference solution is presented, together with its inherent problems and their solution. Results are presented for a variety of situations that include variable fuel, wind and fire breaks. It is found that the equations and their solution efficiently produce simulations for complex problems.
230 citations
TL;DR: A procedure for numerical approximation to two-dimensional, hydraulically-driven fracture propagation in a poroelastic material is described and photographs taken from a high-performance engineering workstation provide insight into the nature of the coupling among the physical phenomena.
Abstract: A procedure for numerical approximation to two-dimensional, hydraulically-driven fracture propagation in a poroelastic material is described. The method uses a partitioned solution procedure to solve a finite element approximation to problems described by the theory of poroelasticity, in conjunction with a finite difference approximation for modelling fluid flow along the fracture. An equilibrium fracture model based on a generalized, Dugdale-Barenblatt concept is used to determine the fracture dimensions. An important feature is that the fracture length is a natural product of the solution algorithm. Two example problems verify the accuracy of the numerical procedure and a third example illustrates a fully-coupled simulation of fracture propagation. Photographs taken from a high-performance engineering workstation provide insight into the nature of the coupling among the physical phenomena.(A)
227 citations
TL;DR: In this article, a frequency-domain approach was proposed to model the wave propagation in complex media for multiple source positions, where solutions for multiple sources are required or when only a limited number of frequency components of the solution are required.
Abstract: The migration, imaging, or inversion of wide-aperture cross-hole data depends on the ability to model wave propagation in complex media for multiple source positions. Computational costs can be considerably reduced in frequency-domain imaging by modeling the frequency-domain steady-state equations, rather than the time-domain equations of motion. I develop a frequency-domain approach in this note that is competitive with time-domain modeling when solutions for multiple sources are required or when only a limited number of frequency components of the solution are required.
213 citations
TL;DR: In this paper, a difference method for the numerical integration of a nonlinear partial integrodifferential equation is considered, where the integral term is treated by means of a convolution quadrature suggested by Lubich.
Abstract: A difference method for the numerical integration of a nonlinear partial integrodifferential equation is considered. The integral term is treated by means of a convolution quadrature suggested by Lubich. Some results from Lubich’s discretized fractional calculus play a crucial role in proving consistency. The verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature. A stability result is derived that is applicable to equations and numerical methods far more general than those treated in this paper.
177 citations
TL;DR: In this article, the steady incompressible Navier-Stokes equations in a 2D driven cavity are solved in primitive variables by means of the multigrid method, where the pressure and the components of the velocity are discretized on staggered grids, a blockimplicit relaxation technique is used to achieve a good convergence and a simplified FMG-FAS algorithm is proposed.
Book•
01 Jan 1990
TL;DR: The Difference Calculus First-Order Difference Equations Linear Difference Equation with Constant Coefficients Linear Partial Different Equations (LPDE) Nonlinear Difference Equational Problems.
Abstract: The Difference Calculus First-Order Difference Equations Linear Difference Equations Linear Difference Equations with Constant Coefficients Linear Partial Difference Equations Nonlinear Difference Equations Problems Appendix Notes and References Bibliography Index
TL;DR: A survey of real analysis can be found in this paper, where the authors present a survey of results from complex analysis in higher dimensions, including linear iterative methods and matrix and vector analysis.
Abstract: Preface to the second edition Preface to the first edition 1. Hyperbolic partial differential equations 2. Analysis of finite difference Schemes 3. Order of accuracy of finite difference schemes 4. Stability for multistep schemes 5. Dissipation and dispersion 6. Parabolic partial differential equations 7. Systems of partial differential equations in higher dimensions 8. Second-order equations 9. Analysis of well-posed and stable problems 10. Convergence estimates for initial value problems 11. Well-posed and stable initial-boundary value problems 12. Elliptic partial differential equations and difference schemes 13. Linear iterative methods 14. The method of steepest descent and the conjugate gradient method Appendix A. Matrix and vectoranalysis Appendix B. A survey of real analysis Appendix C. A Survey of results from complex analysis References Index.
TL;DR: In this article, the authors apply the direct finite difference technique of the incompressible Navier-Stokes equations to unsteady flows around a rectangular cylinder and a circular cylinder as typical building and structure shapes.
TL;DR: In this article, the authors compared the predictive power of the transformed grid and fixed grid methods to resolve the position of the moving phase-change front in a diffusion/convection controlled solidification process.
Abstract: In using finite difference techniques for solving diffusion/ convection controlled solidification processes, the numerical discretization is commonly carried out in one of two ways: (1) transformed grid, in which case the physical space is transformed into a solution space that can be discretized with a fixed grid in space; (2) fixed grid, in which case the physical space is discretized with a fixed uniform orthogonal grid and the effects of the phase change are accounted for on the definition of suitable source terms. In this paper, recently proposed transformed- and fixed-grid methods are outlined. The two methods are evaluated based on solving a problem involving the melting of gallium. Comparisons are made between the predictive power of the two methods to resolve the position of the moving phase-change front
TL;DR: In this article, a variational analysis of a basic state of thermocapillary convection in a cylindrical half-zone of finite length is performed to determine conditions under which the flow will be stable.
Abstract: Energy stability theory has been applied to a basic state of thermocapillary convection occurring in a cylindrical half-zone of finite length to determine conditions under which the flow will be stable. Because of the finite length of the zone, the basic state must be determined numerically. Instead of obtaining stability criteria by solving the related Euler-Lagrange equations, the variational problem is attacked directly by discretization of the integrals in the energy identity using finite differences. Results of the analysis are values of the Marangoni number below which axisymmetric disturbances to the basic state will decay, for various values of the other parameters governing the problem.
TL;DR: These studies show that interfacial friction plays a major role in the unconfined compression response of articular cartilage specimens with small thickness to diameter ratios.
Abstract: A finite element analysis is used to study a previously unresolved issue of the effects of platen-specimen friction on the response of the unconfined compression test; effects of platen permeability are also determined. The finite element formulation is based on the linear KLM biphasic model for articular cartilage and other hydrated soft tissues. A Galerkin weighted residual method is applied to both the solid phase and the fluid phase, and the continuity equation for the intrinsically incompressible binary mixture is introduced via a penalty method. The solid phase displacements and fluid phase velocities are interpolated for each element in terms of unknown nodal values, producing a system of first order differential equations which are solved using a standard numerical finite difference technique. An axisymmetric element of quadrilateral cross-section is developed and applied to the mechanical test problem of a cylindrical specimen of soft tissue in unconfined compression. These studies show that interfacial friction plays a major role in the unconfined compression response of articular cartilage specimens with small thickness to diameter ratios.
TL;DR: The theory of globally convergent homotopy algorithms was introduced in this article, which is a class of algorithms for solving nonlinear systems of equations that are accurate, robust, and converge from an arbitrary starting point almost surely.
Abstract: Probability-one homotopy methods are a class of algorithms for solving nonlinear systems of equations that are accurate, robust, and converge from an arbitrary starting point almost surely. These new globally convergent homotopy techniques have been successfully applied to solve Brouwer fixed point problems, polynomial systems of equations, constrained and unconstrained optimization problems, discretizations of nonlinear two-point boundary value problems based on shooting, finite differences, collocation, and finite elements, and finite difference, collocation, and Galerkin approximations to nonlinear partial differential equations. This paper introduces, in a tutorial fashion, the theory of globally convergent homotopy algorithms, deseribes some computer algorithms and mathematical software, and presents several nontrivial engineering applications.
TL;DR: In this article, a finite volume approach is developed for the solution of the contaminant transport equation in groundwater by defining a triangular control volume over which the dependent variable of the governing equation is averaged, the scheme combines the flexibility in handling complex geometries intrinsic to finite element methods with the simplicity of finite difference techniques.
Abstract: A finite volume approach is developed for the solution of the contaminant transport equation in groundwater. By defining a triangular control volume over which the dependent variable of the governing equation is averaged, the scheme combines the flexibility in handling complex geometries intrinsic to finite element methods with the simplicity of finite difference techniques. High-resolution upwind schemes are employed for the discretization of the advective terms. The technique is based on the concept of “monotone interpolation” to ensure the monotonicity preserving property of the scheme, and on the exact solution of local Riemann problems at the interface between neighboring control volumes. In this way, numerical oscillations are completely avoided for a full range of cell Peclet numbers. Together with the discretization of the dispersive fluxes, an approximation is obtained that is locally first order, but globally of second-order accuracy. As compared to usual upwind schemes, much smaller amounts of numerical viscosity are introduced when sharp concentration fronts occur. A number of numerical tests show good agreement with analytical solutions. A hypothetical problem involving nonequilibrium reaction terms is solved to illustrate the applicability and robustness of the proposed formulation for solving the groundwater transport equations.
TL;DR: In this paper, a theoretical model using the method of characteristics is presented for a Skarstrom-type pressure swing adsorption (PSA) cycle using carbon molecular sieve for the kinetic separation of nitrogen/methane mixtures.
Abstract: A theoretical model using the method of characteristics is presented for a Skarstrom-type pressure swing adsorption (PSA) cycle using carbon molecular sieve for the kinetic separation of nitrogen/methane mixtures. Good agreement with experimental results was obtained. The solution of the reduced nonlinear ordinary differential equations along characteristics, a variable computational grid from the changing bed velocity, and a constant product flow boundary condition are complications that prohibited using the method of characteristics in favor of finite difference or orthogonal collocation methods. These complications have been overcome in the present study to develop a theoretical model, incorporating nonlinear isotherms and a linear driving force rate for diffusion, which requires less computer time and is particularly amenable for studying transient bed dynamics. The effects of feed pressure and concentration, half-cycle time, and product flow rate on kinetic PSA performance are also illustrated.
TL;DR: In this article, the authors discuss numerical difficulties with finite-difference approximations to the thermodynamic conservation laws near sharp, cloud-environment interfaces, and incorporate physical information about condensation-evaporation processes directly into the limiters constraining the antidiffusive fluxes of the FCT methods.
Abstract: We discuss herein numerical difficulties with finite-difference approximations to the thermodynamic conservation laws near sharp, cloud-environment interfaces. The Conservation laws for entropy and water substance variables are coupled through the phase change processes. This coupling of the thermodynamic equations may lead to spurious numerical oscillations that, in general, are not prevented by direct application of traditional monotone methods developed for the uncoupled equations. In order to suppress false oscillations in the solutions, we consider special techniques which derive from the flux-corrected-transport (FCT) methodology. In these, we incorporate physical information about condensation-evaporation processes directly into the limiters constraining the antidiffusive fluxes of the FCT methods. We elaborate upon two different advection-condensation schemes relevant to the two formulations of the advection-condensation problem, commonly used in cloud modeling. For the fractional-time-st...
TL;DR: The finite difference implementation of the parabolic equation method provides a numerical solution to the problem of diffraction of radiowaves by irregular terrain in the presence of atmospheric refraction effects as discussed by the authors.
Abstract: The finite difference implementation of the parabolic equation method provides a numerical solution to the problem of diffraction of radiowaves by irregular terrain in the presence of atmospheric refraction effects. The method has been validated by comparisons with theory and measured data.
TL;DR: In this paper, a third-order finite-difference method was applied to a new three-dimensional, four-phase, equation-of-state, compositional simulator for first-contact miscible flow, waterflooding, immiscible, and multiple-contact condensing gas displacements.
TL;DR: In this article, the authors proposed an optimal polynomial approximation of the Taylor evolution operator of the form eM˜t acting on a vector representing the initial conditions, where M˜ is a spatial operator matrix and t is the time variable.
Abstract: The problem of wave propagation in a linear viscoelastic medium can be described mathematically as an exponential evolution operator of the form eM˜t acting on a vector representing the initial conditions, where M˜ is a spatial operator matrix and t is the time variable. Techniques like finite difference, for instance, are based on a Taylor expansion of this evolution operator. We propose an optimal polynomial approximation of eM˜t based on the powerful method of interpolation in the complex plane, in a domain which includes the eigenvalues of the matrix M˜. The new time‐integration technique is implemented to solve the isotropic viscoacoustic equation of motion. The algorithm is tested for the problem of wave propagation in a homogeneous medium and compared with second‐order temporal differencing and the spectral Chebychev method. The algorithm solves efficiently, and with machine accuracy, the problem of seismic wave propagation with a dissipation mechanism in the sonic band, a characteristic of sedimen...
TL;DR: In this paper, the authors reviewed linear and bilinear interpolation of velocity and introduced a new interpolation scheme using potentiometric head gradients and offered improved accuracy for nonuniform flow in heterogeneous aquifers with abrupt changes in transmissivity.
Abstract: A block-centered, finite difference model of two-dimensional groundwater flow yields velocity values at the midpoints of interfaces between adjacent blocks. Method of characteristics, random walk and particle-tracking models of solute transport require velocities at arbitrary particle locations within the finite difference grid. Particle path lines and travel times are sensitive to the spatial interpolation scheme employed, particularly in heterogeneous aquifers. This paper briefly reviews linear and bilinear interpolation of velocity and introduces a new interpolation scheme. Linear interpolation of velocity is consistent with the numerical solution of the flow equation and preserves discontinuities in velocity caused by abrupt (blocky) changes in transmissivity or hydraulic conductivity. However, linear interpolation yields discontinuous and somewhat unrealistic velocities in homogeneous aquifers. Bilinear interpolation of velocity yields continuous and realistic velocities in homogeneous and smoothly heterogeneous aquifers but does not preserve discontinuities in velocity at abrupt transmissivity boundaries. The new scheme uses potentiometric head gradients and offers improved accuracy for nonuniform flow in heterogeneous aquifers with abrupt changes in transmissivity. The new scheme is equivalent to bilinear interpolation in homogeneous media and is equivalent to linear interpolation where gradients are uniform. Selecting the best interpolation scheme depends, in part, on the conceptualization of aquifer heterogeneity, that is, whether changes in transmissivity occur abruptly or smoothly.
TL;DR: In this paper, the authors presented a method that makes use of the characteristics to suppress spurious oscillations or excessive diffusion near the discontinuities of a thermal wave propagation in a hyperbolic healing conduction problem.
Abstract: For extremely short durations oral very low temperatures (near absolute zero), the classical Fourier heat conduction equation fails and has to be replaced by a hyperbolic equation to account for finite thermal wave propagation. During the last few years, there has been a growing interest in numerical simulation of the hyperbolic heal conduction problem. The schemes used in previous studies were either the classical upwind and central difference or the MacCormack's predictor-corrector schemes. As a result, spurious oscillations or excessive diffusion appeared near the discontinuities. This paper presents a method that makes use of the characteristics to suppress these oscillations or diffusions. The equations governing the hyperbolic heat conduction problem are first transformed into characteristic equations, and the first-order upwind, second-order upwind (Beam-Warming)and second-order central (Lax-Wendroff)schemes are applied based on the direction of the characteristic velocity. It is shown that simple ...
TL;DR: In this paper, a fast quasi-explicit finite difference (FQEFD) method is developed for the simulation of cyclic voltammetry of electrochemical systems comprising coupled heterogeneous and homogeneous kinetics.
TL;DR: In this paper, a three-dimensional heat conduction problem in cartesian coordinate system was solved using the finite element method for predicting the temperature distribution in grain storage bins, which can handle linear and quadratic hexahedron elements with 1, 2, or 3 point Gauss quadrature in each plane.
Abstract: A three-dimensional, heat conduction problem in cartesian coordinate system was solved using the finite element method for predicting the temperature distribution in grain storage bins. The program can handle linear and quadratic hexahedron elements with 1, 2, or 3 point Gauss quadrature in each plane. The model can simulate the temperatures in filled grain bins of any shape and at any location, if the hourly weather data (solar radiation, wind velocity, and ambient air temperature) for the location and the grain temperatures at the start of simulation are available. Other input data required for the model include the three dimensional grid data of a linear or quadratic hexahedron element, and the thermal properties of grain, bin wall material, soil and air. Temperatures predicted by the model were in very good agreement with the measured temperatures in two 5.56 m diameter bins containing rapeseed and barley, respectively, located near Winnipeg. Temperatures predicted by the model in 3.0 m and 4.0 m tall rapeseed bulks of various diameters were compared with the temperatures predicted by 2D finite difference and 3D finite difference models. The temperatures predicted by the 3D finite element model and the 3D finite difference model were nearly identical for different locations in the grain bulks. Three dimensional finite element model predicted higher temperatures by about 5 K to 15 K towards the south side of the bin than the north side, whereas 2D model predicted equal temperatures at these locations.
TL;DR: In this paper, the authors discuss mesh-moving, static mesh-regeneration, and local mesh-refinement algorithms that can be used with a finite difference or finite element scheme to solve initial-boundary value problems for vector systems of time-dependent partial differential equations in two space dimensions and time.
Abstract: We discuss mesh-moving, static mesh-regeneration, and local mesh-refinement algorithms that can be used with a finite difference or finite element scheme to solve initial-boundary value problems for vector systems of time-dependent partial differential equations in two space dimensions and time. A coarse base mesh of quadrilateral cells is moved by an algebraic mesh-movement function so as to follow and isolate spatially distinct phenomena. The local mesh-refinement method recursively divides the time step and spatial cells of the moving base mesh in regions where error indicators are high until a prescribed tolerance is satisfied. The static mesh-regeneration procedure is used to create a new base mesh when the existing one becomes too distorted. The adaptive methods have been combined with a MacCormack finite difference scheme for hyperbolic systems and an error indicator based upon estimates of the local discretization error obtained by Richardson extrapolation. Results are presented for several computational examples.
TL;DR: In this article, the authors present a procedure for extending the useful scope of the finite difference method in solid mechanics applications by evaluating the coefficients of Taylor series expansions for the displacement approximations in terms of rigid body motions, strains and derivatives of strains.
Abstract: Procedures for extending the useful scope of the finite difference method in solid mechanics applications are presented. The improvements centre around the introduction of the physical nature of the deformations into the equations used to formulate the approximate solution. This is accomplished by evaluating the coefficients of Taylor series expansions for the displacement approximations in terms of rigid body motions, strains and derivatives of strains. This procedure is demonstrated with plane stress applications. The ability to interpret the derivative approximations physically allows the fictitious nodes typical of the finite difference method to be rationally incorporated into the model as a means of enforcing traction boundary conditions. An example problem is solved using both regular and irregular meshes. The displacements and stresses compare well with finite element solutions.