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Luis R. Mendez-Nunez and John J. Carroll

Abstract

In a previous paper, the authors discussed the numerical properties of the MacCormack scheme, a finite-difference technique widely used in aerospace simulations. Here the authors report results of its application to the simulation, in two dimensions, of the development of a fully compressible buoyant bubble. The model uses the fully compressible Navier-Stokes equations applied to an inviscid, adiabatic atmosphere. It uses a nonstaggered grid. Both lateral and top boundary conditions are open and essentially reflection-first. The model produced reasonable solutions with no explicit numerical filtering. In regions with locally steep gradients, the MacCormack scheme produces numerical oscillations that locally distort the solution but do not lead to numerical unstability.

These results are compared with those of Droegemeier and of Carpenter et al., who show results using a filtered staggered leapfrog scheme. The fields computed by both schemes are very similar, with those, from the filtered leapfrog being smoother. The major difference is that the speed of propagation of the significant flow features is slower with the leapfrog scheme. The advantage of the MacCormack scheme is that it is numerically stable with no tuned filtering and gives its best results at Courant numbers four times larger than can be used with a leapfrog scheme. In long-term integrations in the presence of very steep gradients, numerically induced oscillations would require some degree of explicit filtering to control these numerical oscillations and improve the quality of the solution. The use of a second-order Fickian filter with the MacCormack scheme weakens the gradients.

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Luis R. Mendez-Nunez and John J. Carroll

Abstract

The MacCormack scheme is a finite-difference scheme widely used in the aerospace simulations. It is a two-step algorithm, and contains a small amount of implicit numerical diffusion that makes it numerically stable without having to use any explicit filtering. It uses a nonstaggered grid. A detailed comparison with the leapfrog and Smolarkiewicz schemes is presented using the nonlinear advection equation and the Euler equations for a variety of conditions at different Courant numbers. Of the schemes tested, the unfiltered leapfrog is the least acceptable for the solution of nonlinear equations. Although it is numerically stable for linear problems, when used to solve nonlinear equations (without using any explicit filtering) it becomes numerically unstable or nonlinearly unstable. Furthermore, it introduces large phase errors, and produces better results with small Courant numbers. The MacCormack scheme is nonlinearly stable, produces modest amounts of numerical dispersion and diffusion, has no phase speed error, and works best at Courant numbers close to 1. When applied to nonlinear equations, the Smolarkiewicz scheme exhibits the least amount of numerical diffusion but more numerical dispersion than the MacCormack scheme. For stability it requires Courant numbers equal to or smaller than 0.5. In practical applications, we recommend the MacCormack scheme for the solution of the nonlinear equations, and either the Smolarkiewicz or the MacCormack scheme for equations involving conservation of a passive scalar.

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