Numerical resolution of Euler equations  through semi-discrete optimal transport

Mirebeau, Jean-Marie

Numerical resolution of Euler equations through semi-discrete optimal transport

Journées équations aux dérivées partielles, (2015), p. 1-16 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé

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].

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/jedp.636

@article{JEDP_2015____A7_0,
     author = {Mirebeau, Jean-Marie},
     title = {Numerical resolution of Euler equations  through semi-discrete optimal transport},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     year = {2015},
     pages = {1-16},
     doi = {10.5802/jedp.636},
     language = {en},
     url = {http://dml.mathdoc.fr/item/JEDP_2015____A7_0}
}

Mirebeau, Jean-Marie. Numerical resolution of Euler equations  through semi-discrete optimal transport. Journées équations aux dérivées partielles,  (2015), pp. 1-16. doi : 10.5802/jedp.636. http://gdmltest.u-ga.fr/item/JEDP_2015____A7_0/

Bibliographie

[1] Abgrall, R Construction of Simple, Stable, and Convergent High Order Schemes for Steady First Order Hamilton–Jacobi Equations, SIAM Journal on Scientific Computing, Tome 31 (2009) no. 4, pp. 2419-2446 | MR 2520283 | Zbl 1197.65167

[2] Aguilera, N E; Morin, P On Convex Functions and the Finite Element Method, SIAM Journal on Numerical Analysis, Tome 47 (2009) no. 4, pp. 3139-3157 | MR 2551161 | Zbl 1204.65076

[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, Tome 31 (2007) no. 4, pp. 497-509 | MR 2372903 | Zbl 1138.49031

[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, Tome 16 (1966) no. 1, pp. 319-361 | Numdam | MR 202082 | Zbl 0148.45301

[5] Aurenhammer, F; Hoffmann, F; Aronov, B Minkowski-Type Theorems and Least-Squares Clustering, Algorithmica, Tome 20 (1998) no. 1, pp. 61-76 | MR 1483422 | Zbl 0895.68135

[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, Tome 40 (2002) no. 1-2, pp. 21-30 | MR 1928314 | Zbl 1058.76586

[7] Benamou, J D; Carlier, G; Cuturi, M; Peyré, G; Nenna, L Iterative Bregman projections for regularized transportation problems, Sci. Comp (2015) | MR 3340204

[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, Tome 44 (2010) no. 4, pp. 737-758 | Numdam | MR 2683581 | Zbl 1192.65138

[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, Tome 260 (2014), pp. 107-126 | MR 3151832

[11] Bernot, M; Figalli, A; Santambrogio, F Generalized solutions for the Euler equations in one and two dimensions, Journal de Mathématiques Pures et Appliquées, Tome 91 (2009) no. 2, pp. 137-155 | MR 2498752 | Zbl 1170.35078

[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 1035755 | Zbl 0687.76016

[13] Brenier, Y The least action principle and the related concept of generalized flows for incompressible perfect fluids, Journal of the American Mathematical Society, Tome 2 (1989) no. 2, pp. 225-255 | MR 969419 | Zbl 0697.76030

[14] Brenier, Y Polar factorization and monotone rearrangement of vector-valued functions, Communications on Pure and Applied Mathematics, Tome 44 (1991) no. 4, pp. 375-417 | MR 1100809 | Zbl 0738.46011

[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 2449787 | Zbl 1143.76386

[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, Tome 10 (1992) no. 3, pp. 255-261 | MR 1186014 | Zbl 0766.58011

[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 3382311

[21] De Goes, F; Breeden, K; Ostromoukhov, V; Desbrun, M Blue noise through optimal transport, ACM Transactions on Graphics (TOG), Tome 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 3340994

[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, Tome 49 (2011) no. 4, pp. 1692-1714 | MR 2831067 | Zbl 1255.65195

[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, Tome 29 (1992) no. 1, pp. 123-135 | MR 1149088 | Zbl 0745.65058

[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, Tome 340 (2005) no. 4, pp. 319-324 | MR 2121899 | Zbl 1067.65119

[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, Tome 177 (2005) no. 2, pp. 151-183 | MR 2188047 | Zbl 1072.49035

[32] Merigot, Q A Multiscale Approach to Optimal Transport, Computer Graphics Forum, Tome 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, Tome 52 (2014) no. 5, pp. 2466-2487 | MR 3268615

[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 $\frac{\partial^{2} z}{\partial x^{2}} \frac{\partial^{2} z}{\partial y^{2}} - {(\frac{\partial^{2} z}{\partial x \partial y})}^{2} = f$ and its discretizations, I, Numerische Mathematik, Tome 54 (1989) no. 3, pp. 271-293 | MR 971703 | Zbl 0659.65116

[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, Tome 4 (1994) no. 5, pp. 586-620 | MR 1296569 | Zbl 0851.76003

[40] Villani, C Optimal transport: Old and new, Springer-Verlag, Berlin (2009), pp. xxii+973 | Article | MR 2459454 | Zbl 1156.53003