Imperial College London


Faculty of Natural SciencesDepartment of Mathematics

Professor of Computational Mathematics



+44 (0)20 7594 3468colin.cotter




755Huxley BuildingSouth Kensington Campus






BibTex format

author = {Budd, CJ and McRae, ATT and Cotter, CJ},
doi = {10.1016/},
journal = {Journal of Computational Physics},
pages = {540--564},
title = {The scaling and skewness of optimally transported meshes on the sphere},
url = {},
volume = {375},
year = {2018}

RIS format (EndNote, RefMan)

AB - In the context of numerical solution of PDEs, dynamic mesh redistribution methods (r-adaptive methods) are an important procedure for increasing the resolution in regions of interest, without modifying the connectivity of the mesh. Key to the success of these methods is that the mesh should be sufficiently refined (locally) and flexible in order to resolve evolving solution features, but at the same time not introduce errors through skewness and lack of regularity. Some state-of-the-art methods are bottom-up in that they attempt to prescribe both the local cell size and the alignment to features of the solution. However, the resulting problem is overdetermined, necessitating a compromise between these conflicting requirements. An alternative approach, described in this paper, is to prescribe only the local cell size and augment this an optimal transport condition to provide global regularity. This leads to a robust and flexible algorithm for generating meshes fitted to an evolving solution, with minimal need for tuning parameters. Of particular interest for geophysical modelling are meshes constructed on the surface of the sphere. The purpose of this paper is to demonstrate that meshes generated on the sphere using this optimal transport approach have good a-priori regularity and that the meshes produced are naturally aligned to various simple features. It is further shown that the sphere's intrinsic curvature leads to more regular meshes than the plane. In addition to these general results, we provide a wide range of examples relevant to practical applications, to showcase the behaviour of optimally transported meshes on the sphere. These range from axisymmetric cases that can be solved analytically to more general examples that are tackled numerically. Evaluation of the singular values and singular vectors of the mesh transformation provides a quantitative measure of the mesh anisotropy, and this is shown to match analytic predictions.
AU - Budd,CJ
AU - McRae,ATT
AU - Cotter,CJ
DO - 10.1016/
EP - 564
PY - 2018///
SN - 0021-9991
SP - 540
TI - The scaling and skewness of optimally transported meshes on the sphere
T2 - Journal of Computational Physics
UR -
UR -
VL - 375
ER -