Three-point Schemes Research Articles

A two-point difference scheme for computing steady-state solutions to the conservative one-dimensional Euler equations

An implicit finite-difference method is presented for obtaining steady-state solutions to the time-dependent, conservative Euler equations for flows containing shocks. The method uses a two-point central-difference scheme for the flux derivatives with dissipation added at supersonic points via the retarded density concept. Application of the method to 1-dimensional nozzle flow equations for various combinations of subsonic and supersonic boundary conditions show the method to be very efficient. Residuals are typically reduced to machine zero in approximately 35 time steps for 50 mesh points. For 1-dimensional Euler calculations, it is shown that the scheme offers two advantages over the more widely-used three-point schemes. The first is in regard to application of boundary conditions, and the second relates to the fact that the two-point algorithm is well-conditioned for large time steps.

Homogeneous difference schemes of a high degree of accuracy on non-uniform nets

The homogeneous difference schemes corresponding to the boundary problems for the differential equation l (p,q,f) u = d dx [ 1 p(x) du dx ] − q (x) u+tf (x) = 0, 0 < x < 1 . (1) M 2 ⩾ p ( x) ⩾ m 1 s> 0, 0 ⩽ q ( x) ⩽ M 3, with piece-wise continuous coefficients ( p, q, f gE Q (0)) include an exact scheme [1], [2]. This scheme enables us to determine the net function which coincides with the exact solution of the boundary problem on an arbitrary non-uniform net s n ( x 0 = 0, x 1, …, x i , …, x N = 1, h i = x i − x i i−1 ). In this article we construct schemes of any (pre-set) order of accuracy on non-homogeneous nets. All the schemes will be of the form l h (p,q,f)y i = 1 kh i δ( ▽y i h ia i h) − D i hy i, + gF i h , kh i = 0,5 ( h i + h i+1 ), Δy i = ▽ y i+1 = y i+1 − y i . (2) The upper index h is the conventional notation for the dependence between the coefficients and the net. In the case of a uniform net ( h i = h = 1 N , i = 1, 2, …, N ) the difference schemes (2) belong to the family of homogeneous three-point conservative schemes [2], [3]. This is also true when the concepts of homogeneity and conservativeness of schemes are generalized to non-uniform nets. Schemes of an increased degree of accuracy prove to be especially useful in a number of cases which are important in practice, such as in the solution of equation (1) with piece-wise constant coefficients with a large number of discontinuities, or in the solution of systems of such equations, or in the solution of equations of heat conduction and diffusion of the form δu dt = L (p, q, f)u .