Fourth‐order finite difference methods for three‐dimensional general linear elliptic problems with variable coefficients

Abstract

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.

Symbolic derivation of finite difference approximations for the three-dimensional Poisson equation

“…Compared to the code for the 7-point scheme, we add 12 new grid points and 6 constraints, and require the polynomial for u to have a higher degree. Eliminate[{eq [1],eq [2],eq [3],eq [4],eq [5],eq [6],eq [7],eq [8],eq [9], eq [10],eq [11],eq [12],eq [13] This 19-point formula is related to the Mehrstellen scheme for the 2D Poisson equation [8] and has been obtained by various authors [1,6]. We show in Section IV that this scheme is stable and achieves fourth-order accuracy.…”

“…We apply a multigrid algorithm [12,13] to two test problems solved in the unit cube Ω = [0,1] 3 with the Dirichlet boundary conditions. The programs were run on a SUN Ultra workstation using FORTRAN 77 programming language in double precision.…”

“…Procedures (based on the truncated Taylor series expansions) for deriving high-order difference approximations for linear partial differential equations have been extensively described in the literature by Gupta, Manohar, Stephenson, and others [1][2][3][4]. These procedures are based upon expressing the solution locally about a given mesh point as a linear combination of the analytic solutions of the differential equation.…”

Closed form numerical solutions of variable coefficient linear second-order elliptic problems

“…Apart from some techniques such as meshless methods [1,2] and those based on particular transformations used to solve special problems [3,4], the most used are related mesh methods as the finite difference method [5,6,7], the finite-volume method [8,9] and the finite element method [10,11].…”

Direct Numerical Simulation of Twin Swirling Flow Jets: Effect of Vortex-Vortex Interaction on Turbulence Modification

“…The fourth order Runge-Kutta schemes [24] are used for time integration. The pressurePoisson equation is solved to obtain the pressure by using the fourth order finite difference method [25].…”