Numerical resolution of Euler equations through semi-discrete optimal transport
Journées équations aux dérivées partielles (2015), article no. 7, 16 p.

Geodesics along the group of volume preserving diffeomorphisms are solutions to Euler equations of inviscid incompressible fluids, as observed by Arnold [4]. On the other hand, the projection onto volume preserving maps amounts to an optimal transport problem, as follows from the generalized polar decomposition of Brenier [14].

We present, in the first section, the framework of semi-discrete optimal transport, initially developed for the study of generalized solutions to optimal transport [1] and now regarded as an efficient approach to computational optimal transport. In a second and largely independent section, we present numerical approaches for Euler equations seen as a boundary value problem [16, 7, 33]: knowing the initial and final positions of some fluid particles, reconstruct intermediate fluid states. Depending on the data, we either recover a classical solution to Euler equations, or a generalized flow [15] for which the fluid particles motion is non-deterministic, as predicted by [39].

DOI: 10.5802/jedp.636
Mirebeau, Jean-Marie 1

1 CNRS et Université Paris-Sud Université Paris-Saclay Départment de Mathématiques Bâtiment 425 91405 Orsay Cedex France
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Mirebeau, Jean-Marie. Numerical resolution of Euler equations  through semi-discrete optimal transport. Journées équations aux dérivées partielles (2015), article  no. 7, 16 p. doi : 10.5802/jedp.636. http://www.numdam.org/articles/10.5802/jedp.636/

[1] Abgrall, R Construction of Simple, Stable, and Convergent High Order Schemes for Steady First Order Hamilton–Jacobi Equations, SIAM Journal on Scientific Computing, Volume 31 (2009) no. 4, pp. 2419-2446 | MR | Zbl

[2] Aguilera, N E; Morin, P On Convex Functions and the Finite Element Method, SIAM Journal on Numerical Analysis, Volume 47 (2009) no. 4, pp. 3139-3157 | MR | Zbl

[3] Ambrosio, L; Figalli, A On the regularity of the pressure field of Brenier’s weak solutions to incompressible Euler equations, Calculus of Variations and Partial Differential Equations, Volume 31 (2007) no. 4, pp. 497-509 | MR | Zbl

[4] Arnold, V Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Annales de l’institut Fourier, Volume 16 (1966) no. 1, pp. 319-361 | Numdam | MR | Zbl

[5] Aurenhammer, F; Hoffmann, F; Aronov, B Minkowski-Type Theorems and Least-Squares Clustering, Algorithmica, Volume 20 (1998) no. 1, pp. 61-76 | MR | Zbl

[6] Benamou, J D; Brenier, Y; Guittet, K The Monge-Kantorovitch mass transfer and its computational fluid mechanics formulation, International Journal for Numerical Methods in Fluids, Volume 40 (2002) no. 1-2, pp. 21-30 | MR | Zbl

[7] Benamou, J D; Carlier, G; Cuturi, M; Peyré, G; Nenna, L Iterative Bregman projections for regularized transportation problems, Sci. Comp, 2015 | MR

[8] Benamou, J-D; Collino, F; Mirebeau, J-M Monotone and Consistent discretization of the Monge-Ampere operator, Mathematics of computation (2015)

[9] Benamou, J-D; Froese, B D; Oberman, A M Two numerical methods for the elliptic Monge-Ampere equation, M2AN. Mathematical Modelling and Numerical Analysis, Volume 44 (2010) no. 4, pp. 737-758 | Numdam | MR | Zbl

[10] Benamou, J-D; Froese, B D; Oberman, A M Numerical solution of the Optimal Transportation problem using the Monge–Ampère equation, Journal of Computational Physics, Volume 260 (2014), pp. 107-126 | MR

[11] Bernot, M; Figalli, A; Santambrogio, F Generalized solutions for the Euler equations in one and two dimensions, Journal de Mathématiques Pures et Appliquées, Volume 91 (2009) no. 2, pp. 137-155 | MR | Zbl

[12] Brenier, Y A combinatorial algorithm for the Euler equations of incompressible flows, Proceedings of the Eighth International Conference on Computing Methods in Applied Sciences and Engineering (Versailles, 1987) (1989), pp. 325-332 | MR | Zbl

[13] Brenier, Y The least action principle and the related concept of generalized flows for incompressible perfect fluids, Journal of the American Mathematical Society, Volume 2 (1989) no. 2, pp. 225-255 | MR | Zbl

[14] Brenier, Y Polar factorization and monotone rearrangement of vector-valued functions, Communications on Pure and Applied Mathematics, Volume 44 (1991) no. 4, pp. 375-417 | MR | Zbl

[15] Brenier, Y The dual least action principle for an ideal, incompressible fluid , Archive for rational mechanics and analysis (1993)

[16] Brenier, Y Generalized solutions and hydrostatic approximation of the Euler equations, Physica D: Nonlinear Phenomena (2008) | MR | Zbl

[17] Bruveris, M; Vialard, François X On Completeness of Groups of Diffeomorphisms (2014) (http://arxiv.org/abs/1403.2089)

[18] Carverhill, A; Pedit, F J Global solutions of the Navier-Stokes equation with strong viscosity, Annals of Global Analysis and Geometry, Volume 10 (1992) no. 3, pp. 255-261 | MR | Zbl

[19] CGAL (http://www.cgal.org/)

[20] De Castro, P M M; Merigot, Q; Thibert, B Intersection of paraboloids and application to Minkowski-type problems, Computational geometry (SoCG’14), ACM, New York, 2014, pp. 308-317 | MR

[21] de Goes, F; Breeden, K; Ostromoukhov, V; Desbrun, M Blue noise through optimal transport, ACM Transactions on Graphics (TOG), Volume 31 (2012) no. 6, pp. 171

[22] de Goes, F; Wallez, C; Huang, J; Zhejiang, U; Pavlov, D Power Particles: An incompressible fluid solver based on power diagrams (2015) (geometry.caltech.edu)

[23] De Lellis, C; Székelyhidi Jr, L The Euler equations as a differential inclusion, Annals of mathematics (2009)

[24] Euler, L Theoria Motus Corporum Solidorum Seu Rigidorum (1765)

[25] Figalli, A; Daneri, S Variational models for the incompressible Euler equations, HCDTE Lecture Notes, Part II (2012) | MR

[26] Froese, B D; Oberman, A M Convergent Finite Difference Solvers for Viscosity Solutions of the Elliptic Monge–Ampère Equation in Dimensions Two and Higher, SIAM Journal on Numerical Analysis, Volume 49 (2011) no. 4, pp. 1692-1714 | MR | Zbl

[27] Kantorovich, L V Kantorovich: On the transfer of masses, Dokl. Akad. Nauk. SSSR, 1942

[28] Kuo, H-J; Trudinger, N S Discrete Methods for Fully Nonlinear Elliptic Equations, SIAM Journal on Numerical Analysis, Volume 29 (1992) no. 1, pp. 123-135 | MR | Zbl

[29] Lévy, B A numerical algorithm for ${L}_{2}$ semi-discrete optimal transport in 3D (2014) (http://arxiv.org/abs/1409.1279v1)

[30] Loeper, G; Rapetti, F Numerical solution of the Monge-Ampère equation by a Newton’s algorithm, Comptes Rendus Mathématique. Académie des Sciences. Paris, Volume 340 (2005) no. 4, pp. 319-324 | MR | Zbl

[31] Ma, X-N; Trudinger, N S; Wang, X-J Regularity of Potential Functions of the Optimal Transportation Problem, Archive for rational mechanics and analysis, Volume 177 (2005) no. 2, pp. 151-183 | MR | Zbl

[32] Merigot, Q A Multiscale Approach to Optimal Transport, Computer Graphics Forum, Volume 30 (2011) no. 5, pp. 1583-1592

[33] Merigot, Q; Mirebeau, J-M Minimal geodesics along volume preserving maps, through semi-discrete optimal transport (2015) (http://arxiv.org/abs/1505.03306)

[34] Merigot, Q; Oudet, E Handling convexity-like constraints in variational problems, SIAM Journal on Numerical Analysis, Volume 52 (2014) no. 5, pp. 2466-2487 | MR

[35] Mirebeau, J-M Adaptive, anisotropic and hierarchical cones of discrete convex functions, Numerische Mathematik (2015), pp. 1-47

[36] Oberman, A M; Ruan, Y An efficient linear programming method for Optimal Transportation (2015) (http://arxiv.org/abs/1509.03668)

[37] Oliker, V I; Prussner, L D On the numerical solution of the equation $\frac{{\partial }^{2}z}{\partial {x}^{2}}\frac{{\partial }^{2}z}{\partial {y}^{2}}-{\left(\frac{{\partial }^{2}z}{\partial x\partial y}\right)}^{2}=f$ and its discretizations, I, Numerische Mathematik, Volume 54 (1989) no. 3, pp. 271-293 | MR | Zbl

[38] Schmitzer, B A sparse algorithm for dense optimal transport, Scale Space and Variational Methods in Computer Vision, Springer International Publishing, Cham, 2015, pp. 629-641

[39] Shnirelman, A I Generalized fluid flows, their approximation and applications, Geometric and Functional Analysis, Volume 4 (1994) no. 5, pp. 586-620 | MR | Zbl

[40] Villani, C Optimal transport: Old and new, Springer-Verlag, Berlin, 2009, pp. xxii+973 | DOI | MR | Zbl

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