Convergence of the finite difference scheme for a general class of the spatial segregation of reaction-diffusion systems
Provides a theoretical convergence guarantee for numerical solutions of a class of reaction-diffusion systems, which is incremental for researchers in numerical analysis.
This work proves that the finite difference scheme for stationary states of a general class of spatial segregation reaction-diffusion systems converges as mesh size tends to zero, assuming the solution components are C^2. The proof also covers convergence for the multi-phase obstacle problem.
In this work we prove convergence of the finite difference scheme for equations of stationary states of a general class of the spatial segregation of reaction-diffusion systems with $m\geq 2$ components. More precisely, we show that the numerical solution $u_h^l$, given by the difference scheme, converges to the $l^{th}$ component $u_l,$ when the mesh size $h$ tends to zero, provided $u_l\in C^2(Ω),$ for every $l=1,2,\dots,m.$ In particular, our proof provides convergence of a difference scheme for the multi-phase obstacle problem.