Development of a Dispersion Relation-Preserving Upwinding Scheme for Incompressible Navier–Stokes Equations on NonStaggered Grids

Abstract

In this article a scheme which preserves the dispersion relation for convective terms is proposed for solving the two-dimensional incompressible Navier-Stokes equations on nonstaggered grids. For the sake of computational efficiency, the splitting methods of Adams-Bashforth and Adams-Moulton are employed in the predictor and corrector steps, respectively, to render second-order temporal accuracy. For the sake of convective stability and dispersive accuracy, the linearized convective terms present in the predictor and corrector steps at different time steps are approximated by a dispersion relation-preserving (DRP) scheme. The DRP upwinding scheme developed within the 13-point stencil framework is rigorously studied by virtue of dispersion and Fourier stability analyses. To validate the proposed method, we investigate several problems that are amenable to exact solutions. Results with good rates of convergence are obtained for both scalar and Navier-Stokes problems.

Three-dimensional bifurcations in a cubic cavity due to buoyancy-driven natural convection

“…For the approximation of the convection terms in flow equations, we will extend the previously proposed two-dimensional advection scheme [28] to carry out the current three-dimensional analysis. The spatial derivative term u x , for example, shown in the momentum equation is approximated as follows:…”

An effective explicit pressure gradient scheme implemented in the two-level non-staggered grids for incompressible Navier–Stokes equations

“…(18), the modified wavenumber range should be sufficient to define a period of sine (or cosine) wave. This explains why the integral range has been chosen to be À p 2 c p 2 [27]. To make E a minimum positive value, the following equation is enforced to achieve the goal:…”

On the development of a triple‐preserving Maxwell's equations solver in non‐staggered grids

“…Discretization of Maxwell's equations remains to approximate the first-order spatial derivative terms shown in Equation (22) to render the algebraic system for equations in (20). One should notice that when discretizing all the first-order derivative terms, shown in (20), in non-staggered grids, care needs to be properly taken of.…”

“…In the approximation of *H x /*x, for example, at a point (i, j), it is essential to take the nodal value of H x | i, j into consideration so as to avoid the so-called even-odd decoupling problem. To enhance numerical stability, one can employ the compact scheme proposed earlier in [21,22]. Referring to Figure 1, *H x /*x| i, j will be approximated by the following equation in a mesh with the grid size of h:…”