Positive and free energy satisfying schemes for diffusion with interaction potentials

Abstract

In this paper, we design and analyze second order positive and free energy satisfying schemes for solving diffusion equations with interaction potentials. The semi-discrete scheme is shown to conserve mass, preserve solution positivity, and satisfy a discrete free energy dissipation law for nonuniform meshes. These properties for the fully-discrete scheme (first order in time) remain preserved without a strict restriction on time steps. For the fully second order (in both time and space) scheme, we use a local scaling limiter to restore solution positivity when necessary. It is proved that such limiter does not destroy the second order accuracy. In addition, these schemes are easy to implement, and efficient in simulations over long time. Both one and two dimensional numerical examples are presented to demonstrate the performance of these schemes.

arXiv comment29 pages, 3 tables, 6 figures

References in corpus (10)

• A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure
• An Energy Stable BDF2 Fourier Pseudo-Spectral Numerical Scheme for the Square Phase Field Crystal Equation
• Convergence Analysis for Second Order Accurate Convex Splitting Schemes for the Periodic Nonlocal Allen-Cahn and Cahn-Hilliard Equations
• A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials
• A free energy satisfying finite difference method for Poisson--Nernst--Planck equations
• Structure preserving schemes for nonlinear Fokker-Planck equations and applications
• A free energy satisfying discontinuous Galerkin method for one-dimensional Poisson--Nernst--Planck systems
• Fisher information regularization schemes for Wasserstein gradient flows
• Semi-dual Regularized Optimal Transport
• Primal dual methods for Wasserstein gradient flows

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• An energy stable and positivity-preserving scheme for the Maxwell-Stefan diffusion system
• A positivity-preserving and energy stable scheme for a quantum diffusion equation