TY - JOUR
T1 - Semi-discrete optimal transport methods for the semi-geostrophic equations
AU - Bourne, David P.
AU - Egan, Charles P.
AU - Pelloni, Beatrice
AU - Wilkinson, Mark
N1 - Funding Information:
DPB would like to thank the UK Engineering and Physical Sciences Research Council (EPSRC) for financial support via the grant EP/R013527/2 Designer Microstructure via Optimal Transport Theory. CPE is supported by The Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by the EPSRC (Grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh. BP and MW gratefully acknowledge the support of the EPSRC via the Grant EP/P011543/1 Analysis of models for large-scale geophysical flows.
Publisher Copyright:
© 2021, The Author(s).
PY - 2022/2
Y1 - 2022/2
N2 - We give a new and constructive proof of the existence of global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations (SG) in geostrophic coordinates, for arbitrary initial measures with compact support. This new proof, based on semi-discrete optimal transport techniques, works by characterising discrete solutions of SG in geostrophic coordinates in terms of trajectories satisfying an ordinary differential equation. It is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations. Using our method, we obtain improved time-regularity for a large class of discrete initial measures, and we compute explicitly two discrete solutions. The method naturally gives rise to an efficient numerical method, which we illustrate by presenting simulations of a 2-dimensional semi-geostrophic flow in geostrophic coordinates generated using a numerical solver for the semi-discrete optimal transport problem coupled with an ordinary differential equation solver.
AB - We give a new and constructive proof of the existence of global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations (SG) in geostrophic coordinates, for arbitrary initial measures with compact support. This new proof, based on semi-discrete optimal transport techniques, works by characterising discrete solutions of SG in geostrophic coordinates in terms of trajectories satisfying an ordinary differential equation. It is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations. Using our method, we obtain improved time-regularity for a large class of discrete initial measures, and we compute explicitly two discrete solutions. The method naturally gives rise to an efficient numerical method, which we illustrate by presenting simulations of a 2-dimensional semi-geostrophic flow in geostrophic coordinates generated using a numerical solver for the semi-discrete optimal transport problem coupled with an ordinary differential equation solver.
KW - semigeostrophic system
KW - atmospheric fluid dynamics
UR - http://www.scopus.com/inward/record.url?scp=85122916295&partnerID=8YFLogxK
U2 - 10.1007/s00526-021-02133-z
DO - 10.1007/s00526-021-02133-z
M3 - Article
SN - 0944-2669
VL - 61
JO - Calculus of Variations and Partial Differential Equations
JF - Calculus of Variations and Partial Differential Equations
IS - 1
M1 - 39
ER -