Implementation of A Posteriori Methods for Enforcing Conservation of Potential Enstrophy and Mass in Discretized Shallow-Water Equations Models

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  • 1 National Research Institute for Mathematical Sciences of the CSIR. Pretoria, South Africa
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Abstract

A Sasaki variational approach is for the first time applied to enforce a posteriori conservation of potential enstrophy and total mass in long-term integrations of two ADI finite-difference approximations of the nonlinear shallow-water equations on a limited-area domain. The performance and accuracy of the variational approach is compared with that of a modified Bayliss-Isaacson a posteriori technique, also designed to enforce conservation of potential enstrophy and total mass, and with that of a periodic application of a two-dimensional high-order Shapiro filter.

While both the variational and the Bayliss-Isaacson a posteriori techniques yielded very satisfactory results after 20 days of numerical integration with regard to conservation of the integral constraints of the shallow-water equations and the accuracy of the solution, the high-order filtering approach performed in a less satisfactory way. This is attributed to the effects of the boundary conditions in the limited-area shallow-water equations models.

The Bayliss-Isaacson technique was found to be more robust and less demanding of CPU time, while the modified Sasaki variational technique is highly dependent on the updating procedure adopted for the Lagrange multiplier. The filtering technique is the most economical in terms of CPU time, but it is inadequate for limited-area domains with non-periodic boundary conditions and coarse meshes. In conclusion further research in this direction is suggested as these techniques provide viable alternatives to the rather complex conserving schemes proposed by other investigators.

Abstract

A Sasaki variational approach is for the first time applied to enforce a posteriori conservation of potential enstrophy and total mass in long-term integrations of two ADI finite-difference approximations of the nonlinear shallow-water equations on a limited-area domain. The performance and accuracy of the variational approach is compared with that of a modified Bayliss-Isaacson a posteriori technique, also designed to enforce conservation of potential enstrophy and total mass, and with that of a periodic application of a two-dimensional high-order Shapiro filter.

While both the variational and the Bayliss-Isaacson a posteriori techniques yielded very satisfactory results after 20 days of numerical integration with regard to conservation of the integral constraints of the shallow-water equations and the accuracy of the solution, the high-order filtering approach performed in a less satisfactory way. This is attributed to the effects of the boundary conditions in the limited-area shallow-water equations models.

The Bayliss-Isaacson technique was found to be more robust and less demanding of CPU time, while the modified Sasaki variational technique is highly dependent on the updating procedure adopted for the Lagrange multiplier. The filtering technique is the most economical in terms of CPU time, but it is inadequate for limited-area domains with non-periodic boundary conditions and coarse meshes. In conclusion further research in this direction is suggested as these techniques provide viable alternatives to the rather complex conserving schemes proposed by other investigators.

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