Abstract
An equation of Monge-Amp\`ere type has, for the first time, been solved numerically on the surface of the sphere in order to generate optimally transported (OT) meshes, equidistributed with respect to a monitor function. Optimal transport generates meshes that keep the same connectivity as the original mesh, making them suitable for r-adaptive simulations, in which the equations of motion can be solved in a moving frame of reference in order to avoid mapping the solution between old and new meshes and to avoid load balancing problems on parallel computers. The semi-implicit solution of the Monge-Amp\`ere type equation involves a new linearisation of the Hessian term, and exponential maps are used to map from old to new meshes on the sphere. The determinant of the Hessian is evaluated as the change in volume between old and new mesh cells, rather than using numerical approximations to the gradients. OT meshes are generated to compare with centroidal Voronoi tesselations on the sphere and are found to have advantages and disadvantages; OT equidistribution is more accurate, the number of iterations to convergence is independent of the mesh size, face skewness is reduced and the connectivity does not change. However anisotropy is higher and the OT meshes are non-orthogonal. It is shown that optimal transport on the sphere leads to meshes that do not tangle. However, tangling can be introduced by numerical errors in calculating the gradient of the mesh potential. Methods for alleviating this problem are explored. Finally, OT meshes are generated using observed precipitation as a monitor function, in order to demonstrate the potential power of the technique.
Highlights
The need to represent scale interactions in weather and climate prediction models has, for many decades, motivated research into the use of adaptive meshes [3, 35, 39]
optimally transported (OT) meshes are generated on the surface of the sphere in order to compare with the centroidal Voronoi meshes generated by Ringler et al [33] using Lloyd’s algorithm
Transported Meshes in Euclidean Geometry Meshes are generated on a finite plane, [−1, 1]2, using the radially symmetric monitor function used by Budd et al [9] defined for each location xi: m (xi) = 1 + α1sech2 α2 R2 − a2 (37)
Summary
The need to represent scale interactions in weather and climate prediction models has, for many decades, motivated research into the use of adaptive meshes [3, 35, 39]. R-adaptivity - mesh redistribution - involves deforming a mesh in order to vary local resolution and was first considered for atmospheric modelling more than twenty years ago by Dietachmayer and Droegemeier [15]. It is an attractive form of adaptivity since it does not involve altering the mesh connectivity, does not create load balancing problems because points are never created or destroyed, does not require mapping of solutions between meshes [27], does not lead to sudden changes in resolution and can be retro-fitted into existing models. We will see that the optimal transport problem on the sphere leads to a slightly different equation of Monge-Ampère type, which has not before been solved numerically on the surface of a sphere, which would be necessary for weather and climate prediction using r-adaptivity
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