Five-point stencil

Numerical Methods for the Navier–Stokes Equations

Published in Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar, Computational Fluid Mechanics and Heat Transfer, 2020

Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar

The use of central differences results in a five-point stencil. The stencil for point (i, j) is split into three components. Stencil components below the diagonal from points (i − 1, j) and (i, j − 1) are collected into the matrix [L] (see Figure 9.2). Stencil components on the diagonal (from points (i, j), (i − 1, j + 1), and (i + 1, j − 1) are collected into matrix [D]. Stencil components above the diagonal (from points (i, j + 1) and (i + 1, j) are collected into matrix [U]. Thus, [A˜] is represented by [L˜]+[D˜]+[U˜], and our matrix–vector representation of Equation 9.81 becomes ([L˜]+[D˜]+[U˜])Δqn+1=−R

Solution of incompressible fluid flow problems with heat transfer by means of an efficient RBF-FD meshless approach

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Journal Information

Published in Numerical Heat Transfer, Part B: Fundamentals, 2019

R. Zamolo, E. Nobile

A localized RBF expansion augmented with a linear polynomial is employed to approximate a generic field ξ in the neighborhood of x: where is the set of the indices j of the n nodes closest to Such local nodes represent the local support for and is the mean point for the local support. When isotropic distributions are employed, we used n = 7 local nodes, since this choice allows a second order accuracy and the higher fraction of positive stencils in the range n < 10 [30], in the case of the Poisson Eq. (10). In the case of Cartesian node arrangements, the traditional five-point stencil is employed, i.e., n = 5.

Five-point stencil

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Numerical Methods for the Navier–Stokes Equations

Solution of incompressible fluid flow problems with heat transfer by means of an efficient RBF-FD meshless approach