Variational Particle Schemes for the Porous Medium Equation and for the System of Isentropic Euler Equations

Abstract

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.

36 pages, 9 figures; re-wrote introduction, added 6 references, added discussion of diagonally implicit Runge-Kutta schemes, moved some material to appendices

Cited by in corpus (10)

• Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation
• On Lagrangian schemes for porous medium type generalized diffusion equations: a discrete energetic variational approach
• Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms
• A Lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes
• Numerical methods for Porous Medium Equation by an Energetic Variational Approach
• Lagrangian schemes for Wasserstein gradient flows
• Numerical Artifacts in the Discontinuous Generalized Porous Medium Equation: How to Avoid Spurious Temporal Oscillations
• High order, semi-implicit, energy stable schemes for gradient flows
• Statistical Mechanics of the Periodic Benjamin-Ono Equation
• A Lipschitz metric for the Camassa--Holm equation