110: 1D Reaction Diffusion equation with two species

(source code) Solve the nonlinear coupled reaction diffusion problem

in $\Omega=(0,1)$ with boundary condition $u_1(0)=1$, $u_2(0)=0$ and $u_1(1)=1$, $u_2(1)=1$.

module Example110_ReactionDiffusion1D_TwoSpecies

using Printf
using VoronoiFVM


function main(;n=100,Plotter=nothing,verbose=false,unknown_storage=:sparse)
    h=1/n
    grid=VoronoiFVM.Grid(collect(0:h:1))


    eps=[1.0,1.0]

    physics=VoronoiFVM.Physics(num_species=2,

                               reaction=function(f,u,node)
                               f[1]=u[1]*u[2]
                               f[2]=-u[1]*u[2]
                               end,

                               flux=function(f,u,edge)
                               nspecies=2
                               f[1]=eps[1]*(u[1,1]-u[1,2])*(0.01+u[2,1]+u[2,2])
                               f[2]=eps[2]*(u[2,1]-u[2,2])*(0.01+u[1,1]+u[1,2])
                               end,

                               source=function(f,node)
                               f[1]=1.0e-4*(0.01+node[1])
                               f[2]=1.0e-4*(0.01+1.0-node[1])
                               end,

                               storage=function(f,u,node)
                               f[1]=u[1]
                               f[2]=u[2]
                               end
                               )

    sys=VoronoiFVM.System(grid,physics,unknown_storage=unknown_storage)

    enable_species!(sys,1,[1])
    enable_species!(sys,2,[1])

    boundary_dirichlet!(sys,1,1,1.0)
    boundary_dirichlet!(sys,1,2,0.0)

    boundary_dirichlet!(sys,2,1,1.0)
    boundary_dirichlet!(sys,2,2,0.0)

    inival=unknowns(sys)
    U=unknowns(sys)
    inival.=0

    control=VoronoiFVM.NewtonControl()
    control.verbose=verbose
    control.damp_initial=0.1
    u5=0
    for xeps in [1.0,0.5,0.25,0.1,0.05,0.025,0.01]
        eps=[xeps,xeps]
        solve!(U,inival,sys,control=control)
        inival.=U
        if isplots(Plotter)
            p=Plotter.plot(grid.coord[1,:],U[1,:], grid=true)
            Plotter.plot!(p,grid.coord[1,:],U[2,:],show=true, title=@sprintf("\$\\varepsilon=%8.3f\$",xeps)),
            Plotter.sleep(0.2)
        end
        u5=U[5]
    end
    return u5
end

function test()
    testval=0.7117546972922056
    main(unknown_storage=:sparse) ≈ testval && main(unknown_storage=:dense) ≈ testval
end
end

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