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Application of the MacCormack Scheme to Atmospheric Nonhydrostatic Models

Luis R. Mendez-NunezDepartment of Land, Air, Water Resources, University of California, Davis, California

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John J. CarrollDepartment of Land, Air, Water Resources, University of California, Davis, California

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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.

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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