SUMMARY We note in this study that the Navier-Stokes equations, when expressed in streamfunction-vorticity fonn, can be approximated to fourth--order accuracy with stencils extending only over a 3 x 3 square of points. The key advantage of the new compact fourth-order scheme is that it allows direct iteration for low~to-mediwn Reynolds numbers. Numerical solutions are obtained for the model problem of the driven cavity and compared with solutions available in the literature. For Re $1500 point-SOR iteration is used and the convergence is fast.

/pdf/a-compact-fourth-order-finite-difference-scheme-for-the-2hdd1pul0e.pdf

A compact fourth‐order finite difference scheme for the steady incompressible Navier‐Stokes equations

High order ADI method for solving unsteady convection-diffusion problems

Two efficient finite difference methods for solving Richards' equation in one dimension are presented, and their use in a range of soils and conditions is investigated. Large time steps are possible when the mass-conserving mixed form of Richards' equation is combined with an implicit iterative scheme, while a hyperbolic sine transform for the matric potential allows large spatial increments even in dry, inhomogeneous soil. Infiltration in a range of soils can be simulated in a few seconds on a personal computer with errors of only a few percent in the amount and distribution of soil water. One of the methods adds points to the space grid as an infiltration or redistribution front advances, thus gaining considerably in efficiency over the other fixed grid method for infiltration problems. In 17-s computing, this advancing front method simulated infiltration, redistribution, and drainage for 50 days in an inhomogeneous soil with nonuniform initial conditions. Only 16 space and 21 time steps were needed for the simulation, which included early ponding with the development and dissipation of a perched water table.

Efficient numerical methods for infiltration using Richards' equation

SUMMARY A new fourth order compact formulation for the steady 2-D incompressible Navier-Stokes equations is presented. The formulation is in the same form of the Navier-Stokes equations such that any numerical method that solve the Navier-Stokes equations can easily be applied to this fourth order compact formulation. In particular in this work the formulation is solved with an efficient numerical method that requires the solution of tridiagonal systems using a fine grid mesh of 601×601. Using this formulation, the steady 2-D incompressible flow in a driven cavity is solved up to Reynolds number of 20,000 with fourth order spatial accuracy. Detailed solutions are presented.

Fourth-order compact formulation of Navier-Stokes equations and driven cavity flow at high Reynolds numbers

Computational Fluid Dynamics: Contents

On propose un schema de differences finies pour l'equation de convection-diffusion a coefficients variables. Le systeme resultant peut etre resolu par des methodes iteratives

A single cell high order scheme for the convection-diffusion equation with variable coefficients

Single cell discretizations of order two and four for biharmonic problems

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

In this article, we give a simple method for developing finite difference schemes on a uniform square gird. We consider a general, two-dimensional, second-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 bnto obtain formulas for gird points near the boundary. In addition to this, these formulas only require nine evaluations for the typical forcing function. Numerical examples are given to demonstrate the performance of one of the fourth-order schemes.

High‐order methods for elliptic equations with variable coefficients

Single Cell High Order Difference Methods for the Helmholtz Equation

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