See here a diffusion example:
You can add reaction to it. It is using the package ParallelStencil.jl which empowers domain scientists to write architecture-agnostic high-level code for parallel high-performance stencil computations on GPUs and CPUs.
In addition, you might want to use implicit time stepping and convergence accelerators. Here you find a presentation of a methodology using the pseudo-transient method and ParallelStencil.jl (and ImplicitGlobalGrid.jl).