A conservative IMEX-Patankar method with flux-based decomposition for positivity-preserving chemotaxis systems

We present a conservative implicit–explicit Patankar-type method for chemotaxis-reaction–diffusion systems modeling tumor-induced angiogenesis. The scheme treats diffusion implicitly via Crank–Nicolson and handles chemotaxis and reaction terms through a Modified Patankar formulation built on a conservative flux-based production–destruction decomposition. We define interfacial fluxes uniquely at cell boundaries, ensuring that influx to one cell equals outflux from its neighbor, thereby preserving discrete mass balance. Both production and destruction terms are weighted by Patankar denominators, yielding a genuine Modified Patankar scheme in the sense of Burchard, Deleersnijder, and Meister. The Patankar stage achieves first-order temporal accuracy; combined with Lie–Trotter splitting, the overall method is first-order in time and second-order in space. Experimental tests confirm positivity preservation under a mild diffusive CFL condition and validate the scheme on a five-component angiogenesis model, demonstrating stability with time steps exceeding explicit limits by two orders of magnitude.