A high-order positivity-preserving local discontinuous Galerkin scheme for Keller–Segel chemotaxis–fluid coupling systems

In this paper, we propose a space and time high-order positivity-preserving local discontinuous Galerkin (PP-LDG) method for the Keller–Segel chemotaxis equations in fluid. Unlike static-media PP-LDG schemes, the scheme proposed in this paper allows the numerical approximation of the Keller–Segel chemotaxis equation to retain its physical meaning even under the influence of non-stationary media (e.g., fluids). Theoretically, we prove (i) optimal spatial convergence rates under sufficient solution regularity and (ii) mass conservation invariance under the positivity-preserving limiter (PPL). In addition, we use the third-order strongly stable preserving Runge–Kutta method for time discretization, which achieves high-precision time approximation while maintaining strong stability. Finally, numerical experiments are presented to verify the necessity of the PPL, as well as the effectiveness and reliability of the proposed method in simulating blow-up solutions and addressing application scenarios.