Standard Central Difference Research Articles

On central difference Approximations to general second order elliptic equations

Consider the second order elliptic equation Lu ≔ − au xx − 2 bu xy − cu yy + d( x, y) u = f( x, y) in [0, 1] × [0, 1], with periodic boundary conditions, and b 2 < ac, a > 0, c > 0, d( x, y) ⩾ 0. Finite difference discretizations require a much stronger condition than ellipticity to give a scheme of positive type. In this paper, it is shown that the standard central difference discretization of (1) is of monotone type although it is not positive type. Specifically, the inverse matrix arising from it has one sign.

A multigrid strongly implicit procedure for transonic potential flowproblems

THE transonic full potential equation in strong conservation form is solved on a body-fitted coordinate system using a strongly implicit procedure enhanced by the multigrid procedure. Weak viscous effects are accounted for by computing the displacement thickness using the NashMcDonald integral boundary-layer equation. The convergence speed of the multigrid procedure is found to be comparable to the state-of-the-art multigrid-alternating direction implicit procedures on identical grids. Several numerical cases, some involving weak viscous effects, are presented and compared with published results. Contents Recently the author and his co-workers developed a relaxation procedure for the solution of the steady transonic full potential equation in a body-fitted coordinate system.1 This procedure has the following steps. The governing equation is linearized at each step of the iteration by lagging the density by one iteration. The density is biased in supersonic regions using the artificial compressibility concept.2'3 Standard central differences are used to discretize the governing equation and the transformation metrics. The above steps lead to the following system of equations:

Calculation of the steady flow through a curved tube using a new finite-difference method

A numerical method is described which is suitable for solving the equations governing the steady motion of a viscous fluid through a slightly curved tube of circular cross-section but which is also applicable to the solution of any problem governed by the steady two-dimensional Navier–Stokes equations in the plane polar co-ordinate system. The governing equations are approximated by a scheme which yields finite-difference equations which are of second-order accuracy with respect to the grid sizes but which have associated matrices which are diagonally dominant. This makes them generally more amenable to solution by iterative techniques than the approximations obtained using standard central differences, while preserving the same order of accuracy.The main object of the investigation is to obtain numerical results for the problem of steady flow through a curved tube which corroborate previous numerical work on this problem in view of a recent paper (Van Dyke 1978) which tends to cast doubt on the accuracy of previous calculations at moderately high values of the Dean number; this is the appropriate Reynolds-number parameter in this problem. The present calculations tend to verify the accuracy of previous results for Dean numbers up to 5000, beyond which it is difficult to obtain accurate results. Calculated properties of the flow are compared with those obtained in previous numerical work, with the predictions of boundary-layer theory for large Dean numbers and with the predictions of Van Dyke (1978).

A finite difference scheme for the incompressible advection-diffusion equation

A representation of continuum space by a second order grid system is proposed. A new finite difference scheme for the two-dimensional incompressible advection-diffusion equation is derived for the model. The difference scheme is flux conserving and contains no spatial truncation error with respect to the model except through approximation of the local velocity. It contains no false diffusion and preserves the sign of positive definite quantities. It is simple to program and is subject only to the usual diffusion and Courant-Friedrichs-Lewy stability conditions. It is stable at all grid mesh sizes or cell Reynolds numbers. A sample one-dimensional problem is presented and comparison made with the standard central difference and “upwind” difference schemes.