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Comparison of Leapfrog, Smolarkiewicz, and MacCormack Schemes Applied to Nonlinear Equations

Luis R. Mendez-NunezAtmospheric Science Section, Department of Land, Air and Water Resources, University of California, Davis, Davis, California

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John J. CarrollAtmospheric Science Section, Department of Land, Air and Water Resources, University of California, Davis, Davis, California

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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.

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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