Positive and free energy satisfying schemes for diffusion with interaction potentials

TL;DR

This paper introduces second order numerical schemes for diffusion equations with interaction potentials that ensure positivity, conserve mass, and dissipate free energy, with proven accuracy and efficiency for long-term simulations.

Contribution

The paper develops and analyzes second order positive and free energy satisfying schemes for diffusion equations with interaction potentials, including a local scaling limiter to maintain positivity.

Findings

Schemes conserve mass and dissipate free energy on nonuniform meshes

Positivity is preserved with a local scaling limiter without losing second order accuracy

Numerical examples demonstrate the schemes' effectiveness in 1D and 2D simulations

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.