Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations
ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 1, pp. 133-166.

Both the porous medium equation and the system of isentropic Euler equations can be considered as steepest descents on suitable manifolds of probability measures in the framework of optimal transport theory. By discretizing these variational characterizations instead of the partial differential equations themselves, we obtain new schemes with remarkable stability properties. We show that they capture successfully the nonlinear features of the flows, such as shocks and rarefaction waves for the isentropic Euler equations. We also show how to design higher order methods for these problems in the optimal transport setting using backward differentiation formula (BDF) multi-step methods or diagonally implicit Runge-Kutta methods.

DOI : 10.1051/m2an/2009043
Classification : 35L65, 49J40, 76M30, 76M28
Mots clés : optimal transport, Wasserstein metric, isentropic Euler equations, porous medium equation, numerical methods
@article{M2AN_2010__44_1_133_0,
     author = {Westdickenberg, Michael and Wilkening, Jon},
     title = {Variational particle schemes for the porous medium equation and for the system of isentropic {Euler} equations},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {133--166},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {1},
     year = {2010},
     doi = {10.1051/m2an/2009043},
     mrnumber = {2647756},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2009043/}
}
TY  - JOUR
AU  - Westdickenberg, Michael
AU  - Wilkening, Jon
TI  - Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2010
SP  - 133
EP  - 166
VL  - 44
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2009043/
DO  - 10.1051/m2an/2009043
LA  - en
ID  - M2AN_2010__44_1_133_0
ER  - 
%0 Journal Article
%A Westdickenberg, Michael
%A Wilkening, Jon
%T Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2010
%P 133-166
%V 44
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2009043/
%R 10.1051/m2an/2009043
%G en
%F M2AN_2010__44_1_133_0
Westdickenberg, Michael; Wilkening, Jon. Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations. ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 1, pp. 133-166. doi : 10.1051/m2an/2009043. http://www.numdam.org/articles/10.1051/m2an/2009043/

[1] L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, Switzerland (2005). | Zbl

[2] V.I. Arnold and B.A. Khesin,Topological methods in hydrodynamics, Applied Mathematical Sciences 125. Springer-Verlag, New York, USA (1998). | Zbl

[3] L.A. Caffarelli, Allocation maps with general cost functions, in Partial differential equations and applications, P. Marcellini, G.G. Talenti and E. Vesintini Eds., Lecture Notes in Pure and Applied Mathematics 177, Marcel Dekker, Inc., New York, USA (1996) 29-35. | Zbl

[4] G.-Q. Chen and D. Wang, The Cauchy problem for the Euler equations for compressible fluids, Handbook of mathematical fluid dynamics I. Elsevier, Amsterdam, North-Holland (2002) 421-543.

[5] C.M. Dafermos, The entropy rate admissibility criterion for solutions of hyperbolic conservation laws. J. Differential Equations 14 (1973) 202-212. | Zbl

[6] W. Gangbo and R.J. Mccann, The geometry of optimal transportation. Acta Math. 177 (1996) 113-161. | Zbl

[7] W. Gangbo and M. Westdickenberg, Optimal transport for the system of isentropic Euler equations. Comm. Partial Diff. Eq. 34 (2009) 1041-1073. | Zbl

[8] E. Hairer, S.P. Norsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems. 2nd edition, Springer, Berlin, Germany (2000). | Zbl

[9] D.D. Holm, J.E. Marsden and T.S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137 (1998) 1-81. | Zbl

[10] http://abel.ee.ucla.edu/cvxopt.

[11] http://www.ziena.com/knitro.htm.

[12] D. Kinderlehrer and N.J. Walkington, Approximation of parabolic equations using the Wasserstein metric. ESAIM: M2AN 33 (1999) 837-852. | Numdam | Zbl

[13] J.E. Marsden and M. West, Discrete mechanics and variational integrators. Acta Numer. 10 (2001) 357-514. | Zbl

[14] J. Nocedal and S.J. Wright, Numerical Optimization. Springer, New York, USA (1999). | Zbl

[15] F. Otto, The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Diff. Eq. 26 (2001) 101-174. | Zbl

[16] J.L. Vázquez, Perspectives in nonlinear diffusion: between analysis, physics and geometry, in International Congress of Mathematicians I (2007) 609-634. | Zbl

[17] C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics 58. American Mathematical Society, Providence, USA (2003). | Zbl

Cité par Sources :