Application of Continuous Dynamic Grid Adaption Techniques to Meteorological Modeling. Part I: Basic Formulation and Accuracy

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  • 1 Bureau of Meteorology Research Centre, Melbourne, Australia
  • | 2 School of meteorology, University of Oklahoma, Norman, Oklahoma
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Abstract

The continuous dynamic grid adaption (CDGA) technique developed in astrophysics and aeronautics is applied, to our knowledge, for the first time to meteorological modeling. The aim of CDGA is to improve the accuracy of numerical solutions of partial differential equations (typically those governing fluid flow) by the use of nonuniform grids that have higher local resolution in regions where the numerical error is presumed to be large. Conceptually, CDGA has some relationship to the well-known technique of grid stretching, but its power lies in its ability to determine an appropriate spatial distribution of grid points automatically and to update this distribution in response to changes in the evolving numerical solution. Application of the technique is facilitated by transforming the governing equations from physical space in which the grid is nonuniform, nonorthogonal and for which the individual grid points are in continuous motion to computational space, which by definition has both a regular and stationary distribution of grid points. The distribution of grid points is found by the solution of “grid-generator” equations, which in turn can be derived as a weighted combination of several variational problems, each of which attempts to enforce a particular desirable property of the grid. These properties include the smoothness and orthogonality of the gridpoint distribution and its response to the user-defined “weight function,” which is a quantitative measure of where the local resolution is to be increased.

The method is applied to several problems of meteorological relevance. The first, Burgers’ equa`tion in one dimension, is used primarily to illustrate the method in a simple context, but also illuminates several features of CDGA, one of which is its ability to improve the accuracy of a numerical solution purely by inducing motion of the grid points. A kinematic frontogenesis problem is used to extend the method to two dimensions, and with the aid of a readily available exact solution, shows the very considerable gains in accuracy that may be achieved over fixed-grid methods. A surprising observation is that the formal order of accuracy of the adaptive results is, for certain parameters, actually greater than for the fixed-grid results. The ability of the technique to allocate multiple zones of high resolution is demonstrated by experiments in which several (four) “cones” are advected by a field of solid-body rotation. The final application is to the evolution of a slab-symmetric thermal in a neutral environment. Again, considerable improvements in accuracy over fixed-grid calculations are achieved, and it is shown that the problem of spurious numerical oscillations associated with rapid variation in an advected field, a problem that has received a great deal of attention in recent times, is greatly alleviated by the CDGA formulation.

Abstract

The continuous dynamic grid adaption (CDGA) technique developed in astrophysics and aeronautics is applied, to our knowledge, for the first time to meteorological modeling. The aim of CDGA is to improve the accuracy of numerical solutions of partial differential equations (typically those governing fluid flow) by the use of nonuniform grids that have higher local resolution in regions where the numerical error is presumed to be large. Conceptually, CDGA has some relationship to the well-known technique of grid stretching, but its power lies in its ability to determine an appropriate spatial distribution of grid points automatically and to update this distribution in response to changes in the evolving numerical solution. Application of the technique is facilitated by transforming the governing equations from physical space in which the grid is nonuniform, nonorthogonal and for which the individual grid points are in continuous motion to computational space, which by definition has both a regular and stationary distribution of grid points. The distribution of grid points is found by the solution of “grid-generator” equations, which in turn can be derived as a weighted combination of several variational problems, each of which attempts to enforce a particular desirable property of the grid. These properties include the smoothness and orthogonality of the gridpoint distribution and its response to the user-defined “weight function,” which is a quantitative measure of where the local resolution is to be increased.

The method is applied to several problems of meteorological relevance. The first, Burgers’ equa`tion in one dimension, is used primarily to illustrate the method in a simple context, but also illuminates several features of CDGA, one of which is its ability to improve the accuracy of a numerical solution purely by inducing motion of the grid points. A kinematic frontogenesis problem is used to extend the method to two dimensions, and with the aid of a readily available exact solution, shows the very considerable gains in accuracy that may be achieved over fixed-grid methods. A surprising observation is that the formal order of accuracy of the adaptive results is, for certain parameters, actually greater than for the fixed-grid results. The ability of the technique to allocate multiple zones of high resolution is demonstrated by experiments in which several (four) “cones” are advected by a field of solid-body rotation. The final application is to the evolution of a slab-symmetric thermal in a neutral environment. Again, considerable improvements in accuracy over fixed-grid calculations are achieved, and it is shown that the problem of spurious numerical oscillations associated with rapid variation in an advected field, a problem that has received a great deal of attention in recent times, is greatly alleviated by the CDGA formulation.

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