Three-point Finite Difference Scheme Research Articles

An optimal time-stepping algorithm for unsteady advection–diffusion problems

We consider the numerical solution of a linear time dependent advection–diffusion problem by an implicit two-weight, three-point finite difference scheme. We extend the scheme proposed by Chadha and Madden (2011), to incorporate an optimal time step selection algorithm for the method. The optimal values of the weights involved in the scheme have been obtained by using the notion of an equivalent differential equation (Warming and Hyett, 1974), and, by enforcing certain constraints due to von Neumann stability and a discrete maximum principle, we obtain sharp bounds for optimal time-stepping in terms of mesh width and problem data. Furthermore, we formulate the time step selection as a root-finding problem, and use this to show the optimal time step exists and is unique. The resulting method, based on optimal values of weights and optimal time-stepping, is of fifth-order in space, and third-order in time. The results of extensive numerical experiments are presented for several classic test problems for various values of the advection and diffusion parameters. These show that our proposed method is superior (quantitatively and qualitatively) to other schemes that can be represented within the framework of two-weighted schemes. We also present comparisons with other high-order methods, and an application of the method to a problem with non-smooth data.

Repeated Spatial Extrapolation: An Extraordinarily Efficient Approach for Option Pricing

A numerical approach for option pricing is proposed which is based on the use of the repeated Richarsdon extrapolation procedure along the price variable. Such a technique, which is applied in conjunction with the classical (centered) three-point finite difference scheme, is tested on two benchmark test-cases, namely the pricing of a European vanilla Call option and the pricing of a European digital Call option under the Black-Scholes model. Even if the final solutions of these problems are non-smooth, the novel extrapolation approach turns out to be extraordinarily efficient. In particular, the option prices can be computed with errors of order 10e-12 or 10e-13 in only some hundredths of second. To the best of our knowledge, these levels of performance are much higher than those that can be achieved using the lattice-based option pricing methods that have been proposed so far.

Uniform boundary stabilization of the finite difference space discretization of the 1−d wave equation

The energy of solutions of the wave equation with a suitable boundary dissipation decays exponentially to zero as time goes to infinity. We consider the finite-difference space semi-discretization scheme and we analyze whether the decay rate is independent of the mesh size. We focus on the one-dimensional case. First we show that the decay rate of the energy of the classical semi-discrete system in which the 1−d Laplacian is replaced by a three-point finite difference scheme is not uniform with respect to the net-spacing size h. Actually, the decay rate tends to zero as h goes to zero. Then we prove that adding a suitable vanishing numerical viscosity term leads to a uniform (with respect to the mesh size) exponential decay of the energy of solutions. This numerical viscosity term damps out the high frequency numerical spurious oscillations while the convergence of the scheme towards the original damped wave equation is kept. Our method of proof relies essentially on discrete multiplier techniques.

On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids

AbstractWe consider finite difference approximations of solutions of inverse Sturm‐Liouville problems in bounded intervals. Using three‐point finite difference schemes, we discretize the equations on so‐called optimal grids constructed as follows: For a staggered grid with 2 k points, we ask that the finite difference operator (a k × k Jacobi matrix) and the Sturm‐Liouville differential operator share the k lowest eigenvalues and the values of the orthonormal eigenfunctions at one end of the interval. This requirement determines uniquely the entries in the Jacobi matrix, which are grid cell averages of the coefficients in the continuum problem. If these coefficients are known, we can find the grid, which we call optimal because it gives, by design, a finite difference operator with a prescribed spectral measure. We focus attention on the inverse problem, where neither the coefficients nor the grid are known.A key question in inversion is how to parametrize the coefficients, i.e., how to choose the grid. It is clear that, to be successful, this grid must be close to the optimal one, which is unknown. Fortunately, as we show here, the grid dependence on the unknown coefficients is weak, so the inversion can be done on a precomputed grid for an a priori guess of the unknown coefficients. This observation leads to a simple yet efficient inversion algorithm, which gives coefficients that converge pointwise to the true solution as the number k of data points tends to infinity. The cornerstone of our convergence proof is showing that optimal grids provide an implicit, natural regularization of the inverse problem, by giving reconstructions with uniformly bounded total variation. The analysis is based on a novel, explicit perturbation analysis of Lanczos recursions and on a discrete Gel'fand‐Levitan formulation. © 2005 Wiley Periodicals, Inc.

Subsampling of fine features in finite-difference frequency-domain simulations

An improved three-point finite-difference scheme is presented, which deals correctly with all terms up to 2nd order in the expansion of the derivative, regardless of the existence of multiple dielectric interfaces between sample points. The improved differencing scheme is implemented into a 1D slab-mode solver for validation of accuracy and a 2D mode solver. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 44: 95–101, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20558

Three-point finite-difference schemes, Padé and the spectral Galerkin method. I. One-sided impedance approximation

A method for calculating special grid placement for three-point schemes which yields exponential superconvergence of the Neumann to Dirich-let map has been suggested earlier. Here we show that such a grid placement can yield impedance which is equivalent to that of a spectral Galerkin method, or more generally to that of a spectral Galerkin-Petrov method. In fact we show that. for every stable Galerkin-Petrov method there is a three-point scheme which yields the same solution at the boundary. We discuss the application of this result to partial differential equations and give numerical examples. We also show equivalence at one corner of a two-dimensional optimal grid with a spectral Galerkin method.

ON ASYMPTOTIC ERROR EXPANSIONS FOR SINGULAR BOUNDARY VALUE PROBLEMS

The existence of an asymptotic expansion for the global error of a standard three-point finite difference scheme applied to solve BVPs for ordinary differential equations of second order with a singularity of the first kind is considered. It turns out that even for very smooth problem data such asymptotic expansion does not exist, in general. More precisely, its length is restricted by unbounded inhomogeneities occurring in variational equations due to the singularity.

Optimal finite difference grids and rational approximations of the square root I. Elliptic problems

The main objective of this paper is optimization of second-order finite difference schemes for elliptic equations, in particular, for equations with singular solutions and exterior problems. A model problem corresponding to the Laplace equation on a semi-infinite strip is considered. The boundary impedance (Neumann-to-Dirichlet map) is computed as the square root of an operator using the standard three-point finite difference scheme with optimally chosen variable steps. The finite difference approximation of the boundary impedance for data of given smoothness is the problem of rational approximation of the square root on the operator's spectrum. We have implemented Zolotarev's optimal rational approx-imant obtained in terms of elliptic functions. We have also found that a geometrical progression of the grid steps with optimally chosen parameters is almost as good as the optimal approximant. For bounded operators it increases from second to exponential the convergence order of the finite difference impedance with the convergence rate proportional to the inverse of the logarithm of the condition number. For the case of unbounded operators in Sobolev spaces associated with elliptic equations, the error decays as the exponential of the square root of the mesh dimension. As an example, we numerically compute the Green function on the boundary for the Laplace equation. Some features of the optimal grid obtained for the Laplace equation remain valid for more general elliptic problems with variable coefficients. © 2000 John Wiley & Sons, Inc.

Determination of electronic chemical potential within density functional theory

The Donnelly–Parr version of the density-matrix functional theory [R.A. Donnelly, R.G. Parr, J. Chem. Phys. 69 (1978) 4431] is tested with various approximate energy functionals being used. The results obtained are discussed in the context of the functional N-representability condition. It has been demonstrated that the Euler–Lagrange equation proposed by Donnelly and Parr has no solutions within the set of N-representable density matrices. It follows that the validity of Sanderson's Principle of Electronegativity Equalization is open to question. Techniques for evaluating the electronic chemical potential within density functional theory are briefly discussed. The approximation of the electronic chemical potential with the three-point finite-difference scheme seems to be the most relevant route to its evaluation at the present stage of development.

Optimal finite difference grids and rational approximations of the square root I. Elliptic problems

The main objective of this paper is optimization of second-order finite difference schemes for elliptic equations, in particular, for equations with singular solutions and exterior problems. A model problem corresponding to the Laplace equation on a semi-infinite strip is considered. The boundary impedance (Neumann-to-Dirichlet map) is computed as the square root of an operator using the standard three-point finite difference scheme with optimally chosen variable steps. The finite difference approximation of the boundary impedance for data of given smoothness is the problem of rational approximation of the square root on the operator's spectrum. We have implemented Zolotarev's optimal rational approx-imant obtained in terms of elliptic functions. We have also found that a geometrical progression of the grid steps with optimally chosen parameters is almost as good as the optimal approximant. For bounded operators it increases from second to exponential the convergence order of the finite difference impedance with the convergence rate proportional to the inverse of the logarithm of the condition number. For the case of unbounded operators in Sobolev spaces associated with elliptic equations, the error decays as the exponential of the square root of the mesh dimension. As an example, we numerically compute the Green function on the boundary for the Laplace equation. Some features of the optimal grid obtained for the Laplace equation remain valid for more general elliptic problems with variable coefficients. © 2000 John Wiley & Sons, Inc.

Analysis of MOS Device Capacitance-Voltage Characteristics Based on the Self-Consistent Solution of the Schrödinger and Poisson Equations

ABSTRACTThe one-dimensional Schridinger and Poisson equations have been numerically solved in metal-oxide-semiconductor devices using a three-point finite difference scheme with a non-uniform mesh size. The capacitance-voltage characteristic of the structure has been calculated via this self-consistent approach and results have been compared with data obtained from the resolution of Poisson equation using different approximated methods based on the Boltzmann statistic with and without a first order quantum effect correction or the exact Fermi-Dirac statistic. The present work permits to evaluate and quantify the errors made by these approximations in determining the thickness of ultra-thin oxides.

Streamline Upwind Finite-Element Method for Overland Flow

A streamline upwind, finite-element method is proposed to solve the two-dimensional (2D) kinematic wave and shallow water wave equations for overland flow problems. The spatial domain is discretized into a network of quadrilateral, bilinear finite elements, and time derivatives are treated by a three-point finite-difference scheme to increase numerical stability. Solutions computed by the streamline upwind method are compared to those of the Galerkin method. Streamline upwind and Galerkin solutions of the kinematic wave equation exhibited good agreement with an analytic solution for flow over an inverted cone. However, the Galerkin solution exhibited oscillations during the steady portion of the runoff hydrograph. Galerkin solutions of the shallow water wave equations for inverted cone and bowl-shaped surfaces displayed oscillations in time and space, while the streamline upwind method exhibited no oscillatory behavior. Results generated by the streamline upwind method agreed well with experimental data sets and simulations conducted with the MacCormack finite-difference method.

High-order compact schemes for second-order ordinary differential equations with singular coefficients

Homogeneous three-point finite-difference schemes of arbitrary order of accuracy for a second-order Dirichlet problem with generalized coefficients and generalized right-hand side are constructed and validated. Bibliography: 8 titles.

High Order Difference Schemes for Unsteady One-Dimensional Diffusion-Convection Problems

For unsteady 1D diffusion-convection problems, this paper develops an extensive analysis of two-level three-point finite difference schemes of order 2 in time and 4 in space. This general class of FDS includes several schemes independently proposed by different authors. One main objective is the identification of difference schemes yielding satisfactory numerical results for strongly convective problems (i.e., when the cell Reynolds number bα = λ h/2 is greater than unity). The stability and the oscillatory behaviour of the schemes are carefully studied and the analyses are completed by some numerical experiments. We outline some key points: (i) the great difficulty to obtain accurate numerical results for large values of α (ii) the possibility of virtually optimum schemes is essentially theoretical and requires, in practice, careful experiments; (iii) for strongly convective problems, some second-order explicit schemes are almost as efficient (and less costly) than implicit fourth-order schemes.

An Investigation of the Transient Thermal Analysis of Spur Gears

A finite element computer program has been developed for evaluating the transient behavior of surface temperature in high performance spur gears. The program is an extension of a previously developed code that was limited to steady-state behavior. The time dimension is implemented using two- and three-point finite difference schemes. The different schemes are provided to the user for the purpose of numerical stability and convergence studies. A detailed explanation of the gear cooling process leading to the establishment of a modified Blok model is also included. Other conventional models for approximating the heat transfer coefficients are available for comparison. Preliminary results are given showing snap shots of gear temperature contours at the initial stages of tooth engagement.

A comparison of finite element and finite difference solutions of the one- and two-dimensional Burgers' equations

Solutions to the one- and two-dimensional Burgers' equations with moderate to severe internal and boundary gradients have been used to compare minimum truncation error three, five-, and seven-point finite difference schemes with linear, quadratic, and cubic rectangular finite element schemes. The various schemes demonstrate the theoretically predicted convergence rates if the mesh is sufficiently refined. In one dimension the quadratic finite element scheme and the five-point finite difference scheme are computationally the most efficient. In multidimensions the finite element method is less economical than the finite difference method even if the group representation is used for the convective terms. In two dimensions the linear group finite element representation and the three-point finite difference scheme are computationally the most efficient on a coarse mesh or if a severe gradient is present.

Nodal Equations Derived from Invariant Imbedding Theory

In one-dimensional systems which consist of N nodes, the two N response matrix equations for the partial currents through the node interfaces have been transformed into a set of N three-point equations with the total in-current per node as the new variable. The resulting coefficients which describe the coupling between neighboring nodes are expressed in terms of the reflection and transmission matrices of the invariant imbedding theory. These coupling coefficients can be compared with those of other nodal equations. In the case of slab geometry this has been illustrated by a direct comparison with the familiar finite difference formulation with the average flux per node as the dependent variable. Also the relation between the method presented here and the so-called rigorous finite difference equations has been established. The advantage of this method lies in the fact that the flexibility of the response matrix methods—which describe the nodes in terms of invariant imbedding concepts—has been condensed into the conventional three-point finite difference scheme, for which many well-established solution methods exist.