A unified structure preserving scheme for a multi-species model with a gradient flow structure and nonlocal interactions via singular kernels

TL;DR

This paper introduces a semi-implicit finite volume scheme for a multi-species ionic fluid model with nonlocal interactions, ensuring positivity, mass conservation, and energy dissipation, while efficiently handling singular kernels.

Contribution

It presents a novel second-order accurate scheme with a fast convolution algorithm for singular kernels, along with rigorous error estimates and extensive numerical validation.

Findings

The scheme is unconditionally stable and energy dissipative.

Numerical experiments confirm the scheme's convergence and efficiency.

Applications include modeling ion concentration and blowup phenomena.

Abstract

In this paper, we consider a nonlinear and nonlocal parabolic model for multi-species ionic fluids and introduce a semi-implicit finite volume scheme, which is second order accurate in space, first order in time and satisfies the following properties: positivity preserving, mass conservation and energy dissipation. Besides, our scheme involves a fast algorithm on the convolution terms with singular but integrable kernels, which otherwise impedes the accuracy and efficiency of the whole scheme. Error estimates on the fast convolution algorithm are shown next. Numerous numerical tests are provided to demonstrate the properties, such as unconditional stability, order of convergence, energy dissipation and the complexity of the fast convolution algorithm. Furthermore, extensive numerical experiments are carried out to explore the modeling effects in specific examples, such as, the steric repulsion, the concentration of ions at the boundary and the blowup phenomenon of the Keller-Segel equations.