Positive And Free Energy Satisfying Schemes For Diffusion With Interaction Potentials
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, a local scaling limiter is introduced 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. Both one and two dimensional numerical examples are presented to demonstrate the performance of these schemes.
H. Liu and W. Maimaitiyiming, "Positive And Free Energy Satisfying Schemes For Diffusion With Interaction Potentials," Journal of Computational Physics, vol. 419, article no. 109483, Elsevier, Oct 2020.
The definitive version is available at https://doi.org/10.1016/j.jcp.2020.109483
Mathematics and Statistics
Keywords and Phrases
Drift-diffusion equations; Energy dissipation; Implicit-explicit scheme; Positivity preserving
International Standard Serial Number (ISSN)
Article - Journal
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15 Oct 2020