Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations

TL;DR

This paper introduces a semi-discrete scheme for 2D Keller-Segel equations that is positivity-preserving, asymptotic-preserving, unconditionally stable under certain conditions, and suitable for challenging simulations.

Contribution

The paper presents a novel symmetrization-based semi-discrete scheme that avoids nonlinear solvers and maintains key physical properties for 2D Keller-Segel equations.

Findings

Unconditionally stable for initial conditions below a threshold

Preserves positivity and conserves mass in simulations

Demonstrates superior performance in numerical tests

Abstract

We propose a semi-discrete scheme for 2D Keller-Segel equations based on a symmetrization reformation, which is equivalent to the convex splitting method and is free of any nonlinear solver. We show that, this new scheme is unconditionally stable as long as the initial condition does not exceed certain threshold, and it asymptotically preserves the quasi-static limit in the transient regime. Furthermore, we prove that the fully discrete scheme is conservative and positivity preserving, which makes it ideal for simulations. The analogical schemes for the radial symmetric cases and the subcritical degenerate cases are also presented and analyzed. With extensive numerical tests, we verify the claimed properties of the methods and demonstrate their superiority in various challenging applications.