A higher-order gradient flow scheme for a singular one-dimensional\n diffusion equation

2015/09/01 by Bertram Düring, Düring, Bertram, Philipp Fuchs +3
Engineering · Mathematics · #35K20 #65M99 #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1509.00384

openalex publication_date 2015/09/01 · openalex created_date 2022/09/06 · openalex updated_date 2026/08/01

Abstract

A nonlinear diffusion equation, interpreted as a Wasserstein gradient flow,\nis numerically solved in one space dimension using a higher-order minimizing\nmovement scheme based on the BDF (backward differentiation formula)\ndiscretization. In each time step, the approximation is obtained as the\nsolution of a constrained quadratic minimization problem on a\nfinite-dimensional space consisting of piecewise quadratic basis functions. The\nnumerical scheme conserves the mass and dissipates the G-norm of the two-step\nBDF time approximation. Numerically, also the discrete entropy and variance are\ndecaying. The decay turns out to be exponential in all cases. The corresponding\ndecay rates are computed numerically for various grid numbers.\n

Citations

• Long-time behaviour of a fully discrete Lagrangian scheme for a family of fourth order
• Geodesic convexity of the relative entropy in reversible Markov chains
• Entropic Wasserstein Gradient Flows

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