In this article, we give a simple method for deriving finite difference schemes on a uniform cubic grid. We consider a general, three-dimensional, second-order, linear, elliptic partial differential equation with variable coefficients. We derive two simple fourthorder schemes. When the coefficients of the second-order mixed derivatives are equal to zero, the fourth-order scheme requires only 19 grid points. When the coefficients of the mixed derivatives are not equal to zero and the coefficients of U,, U,, and U, are equal, we require the 27 points of the cubic grid. Numerical examples are given to demonstrate the performance of the two schemes derived. There does not exist a fourthorder scheme involving 27 grid points for the general case.
In this article, we give a simple method for developing finite difference schemes on a uniform square grid. We consider a general, two-dimensional, sezond-order, partial differential equation with variable coefficients. In the case of a nine-point scheme, we obtain the known results of Young and Dauwalder in a fairly elegant fashion. We show how this can be extended to obtain fourth-order schemes on thirteen points. We derive two such schemes which are attractive because they can be adapted quite easily to obtain formulas for grid points near the boundary. In addition to this, these formulas only require nine evaluations for the forcing function. Numerical examples are given to demonstrate the typical performance of one of the fourth-order schemes.
Abstract. In this paper /'-stable methods of 0(hb) and 0(/i8) with minimal phase-lag (frequency distortion) are derived. Numerical results for both linear and nonlinear problems are presented.
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