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.
@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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {44},
year = {2010},
pages = {133-166},
doi = {10.1051/m2an/2009043},
mrnumber = {2647756},
zbl = {pre05693768},
language = {en},
url = {http://dml.mathdoc.fr/item/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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 44 (2010) pp. 133-166. doi : 10.1051/m2an/2009043. http://gdmltest.u-ga.fr/item/M2AN_2010__44_1_133_0/
[1] , and , Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, Switzerland (2005). | Zbl 1145.35001
[2] and ,Topological methods in hydrodynamics, Applied Mathematical Sciences 125. Springer-Verlag, New York, USA (1998). | Zbl 0902.76001
[3] , 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 0883.49030
[4] and , The Cauchy problem for the Euler equations for compressible fluids, Handbook of mathematical fluid dynamics I. Elsevier, Amsterdam, North-Holland (2002) 421-543. | Zbl pre01942876
[5] , The entropy rate admissibility criterion for solutions of hyperbolic conservation laws. J. Differential Equations 14 (1973) 202-212. | Zbl 0262.35038
[6] and , The geometry of optimal transportation. Acta Math. 177 (1996) 113-161. | Zbl 0887.49017
[7] and , Optimal transport for the system of isentropic Euler equations. Comm. Partial Diff. Eq. 34 (2009) 1041-1073. | Zbl 1182.35161
[8] , and , Solving Ordinary Differential Equations I: Nonstiff Problems. 2nd edition, Springer, Berlin, Germany (2000). | Zbl 0789.65048
[9] , and , The Euler-Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137 (1998) 1-81. | Zbl 0951.37020
[10] http://abel.ee.ucla.edu/cvxopt.
[11] http://www.ziena.com/knitro.htm.
[12] and , Approximation of parabolic equations using the Wasserstein metric. ESAIM: M2AN 33 (1999) 837-852. | Numdam | Zbl 0936.65121
[13] and , Discrete mechanics and variational integrators. Acta Numer. 10 (2001) 357-514. | Zbl 1123.37327
[14] and , Numerical Optimization. Springer, New York, USA (1999). | Zbl 1104.65059
[15] , The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Diff. Eq. 26 (2001) 101-174. | Zbl 0984.35089
[16] , Perspectives in nonlinear diffusion: between analysis, physics and geometry, in International Congress of Mathematicians I (2007) 609-634. | Zbl 1128.35060
[17] , Topics in optimal transportation, Graduate Studies in Mathematics 58. American Mathematical Society, Providence, USA (2003). | Zbl 1106.90001