Difference Schemes For Computation Research Articles

Computer difference scheme for a singularly perturbed elliptic convection–diffusion equation in the presence of perturbations

A grid approximation of a boundary value problem for a singularly perturbed elliptic convection–diffusion equation with a perturbation parameter e, e ∈ (0,1], multiplying the highest order derivatives is considered on a rectangle. The stability of a standard difference scheme based on monotone approximations of the problem on a uniform grid is analyzed, and the behavior of discrete solutions in the presence of perturbations is examined. With an increase in the number of grid nodes, this scheme does not converge -uniformly in the maximum norm, but only conditional convergence takes place. When the solution of the difference scheme converges, which occurs if N 1 -1 N 2 -1 ≪ e, where N 1 and N 2 are the numbers of grid intervals in x and y, respectively, the scheme is not -uniformly well-conditioned or e-uniformly stable to data perturbations in the grid problem and to computer perturbations. For the standard difference scheme in the presence of data perturbations in the grid problem and/or computer perturbations, conditions imposed on the “parameters” of the difference scheme and of the computer (namely, on e, N 1,N 2, admissible data perturbations in the grid problem, and admissible computer perturbations) are obtained that ensure the convergence of the perturbed solutions as N 1,N 2 → ∞, e ∈ (0,1]. The difference schemes constructed in the presence of the indicated perturbations that converges as N 1,N 2 → ∞ for fixed e, e ∈ (0,1, is called a computer difference scheme. Schemes converging e-uniformly and conditionally converging computer schemes are referred to as reliable schemes. Conditions on the data perturbations in the standard difference scheme and on computer perturbations are also obtained under which the convergence rate of the solution to the computer difference scheme has the same order as the solution of the standard difference scheme in the absence of perturbations. Due to this property of its solutions, the computer difference scheme can be effectively used in practical computations.

Standard finite difference scheme for a singularly perturbed elliptic convection–diffusion equation on a rectangle under computer perturbations

A singularly perturbed elliptic convection–diffusion equation with a perturbation parameter e (e ∈ (0, 1]) is considered on a rectangle. As applied to this equation, a standard finite difference scheme on a uniform grid is studied under computer perturbations. This scheme is not e-uniformly stable with respect to perturbations. The conditions imposed on a “computing system” are established under which a converging standard scheme (referred to as a computer difference scheme) remains stable.

Computer Difference Scheme for a Singularly Perturbed Reaction- Diffusion Equation in the Presence of Perturbations

In this paper, for a singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter e 2 , e ∈ (0,1], multiplying the highest-order derivative in the equation, an initial-boundary value Dirichlet problem is considered. For this problem, a standard difference scheme constructed by using monotone grid approximations of the differential problem on uniform grids, is studied in the presence of computer perturbations. Perturbations of grid solutions are studied, which are generated by computer perturbations, i.e., the computations on a computer. The conditions imposed on admissible computer perturbations are obtained under which the accuracy of the perturbed computer solution is the same by order as the solution of an unperturbed difference scheme, i.e., a standard scheme in the absence of perturbations. The schemes of this type with controlled computer perturbations belong to computer difference schemes, also named reliable difference schemes.

Computer difference scheme for a singularly perturbed convection-diffusion equation

The Dirichlet problem for a singularly perturbed ordinary differential convection-diffusion equation with a perturbation parameter ɛ (that takes arbitrary values from the half-open interval (0, 1]) is considered. For this problem, an approach to the construction of a numerical method based on a standard difference scheme on uniform meshes is developed in the case when the data of the grid problem include perturbations and additional perturbations are introduced in the course of the computations on a computer. In the absence of perturbations, the standard difference scheme converges at an \(\mathcal{O}\)(δst) rate, where δst = (ɛ + N−1)−1N−1 and N + 1 is the number of grid nodes; the scheme is not ɛ-uniformly well conditioned or stable to perturbations of the data. Even if the convergence of the standard scheme is theoretically proved, the actual accuracy of the computed solution in the presence of perturbations degrades with decreasing ɛ down to its complete loss for small ɛ (namely, for ɛ = \(\mathcal{O}\)(δ−2maxi,j|δaij| + δ−1 maxi, j|δbij|), where δ = δst and δaij, δbij are the perturbations in the coefficients multiplying the second and first derivatives). For the boundary value problem, we construct a computer difference scheme, i.e., a computing system that consists of a standard scheme on a uniform mesh in the presence of controlled perturbations in the grid problem data and a hypothetical computer with controlled computer perturbations. The conditions on admissible perturbations in the grid problem data and on admissible computer perturbations are obtained under which the computer difference scheme converges in the maximum norm for ɛ ∈ (0, 1] at the same rate as the standard scheme in the absence of perturbations.

A new family of (5,5)CC-4OC schemes applicable for unsteady Navier–Stokes equations

In this paper, a new family of implicit compact finite difference schemes for computation of unsteady convection–diffusion equation with variable convection coefficient is proposed. The schemes which are fourth order accurate in space and second or lower order accurate in time depending on the choice of weighted time average parameter are then applied to unsteady Navier–Stokes system. The proposed schemes, where transport variable and its first derivatives are carried as the unknowns, combine virtues of compact discretization and Padé scheme for spatial derivative. These schemes which are based on a five point stencil with constant coefficients, named as “(5,5) Constant Coefficient 4th Order Compact” [(5,5)CC-4OC], give rise to a diagonally dominant system of equations and shows higher accuracy and better phase and amplitude error characteristics than some of the standard methods. These schemes are capable of using a grid aspect ratio other than unity and are unconditionally stable. They efficiently capture both transient and steady solutions of linear and nonlinear convection–diffusion equations with Dirichlet as well as Neumann boundary conditions. Subsequently the proposed schemes are applied to problems governed by the incompressible Navier–Stokes equations. The results obtained are in excellent agreement with analytical and available numerical results in all cases, establishing efficiency and accuracy of the proposed schemes.

A Finite Difference Model for Valuation of Employee Stock Options

Employee stock option grants are a common incentive for employees and are a key remuneration device. These options differ from ordinary options in that they cannot be traded nor hedged. Nevertheless, the work of Carpenter enables one to price these options within a Black-Scholes framework, with one additional parameter calibrated from historical data. We develop a model where the price of the grant obeys the Black-Scholes differential equation with two additional parameters: one which controls the rate at which employees forfeit unvested options, and another which controls the rate at which employees exercise vested options. We implement a finite difference scheme for computation of the option values derived from this model.