406: 1D Weird Surface Reaction

(source code)

Species $A$ and $B$ exist in the interior of the domain, species $C$ lives a the boundary $\Gamma_1$. We assume a heterogeneous reaction scheme where $A$ reacts to $B$ with a rate depending on $\nabla A$ near the surface

\[\begin{aligned} A &\leftrightarrow B\\ \end{aligned}\]

In $\Omega$, both $A$ and $B$ are transported through diffusion:

\[\begin{aligned} \partial_t u_B - \nabla\cdot D_A \nabla u_A & = f_A\\ \partial_t u_B - \nabla\cdot D_B \nabla u_B & = 0\\ \end{aligned}\]

Here, $f(x)$ is a source term creating $A$. On $\Gamma_2$, we set boundary conditions

\[\begin{aligned} D_A \nabla u_A & = 0\\ u_B&=0 \end{aligned}\]

describing no normal flux for $A$ and zero concentration of $B$. On $\Gamma_1$, we use the mass action law to describe the boundary reaction and the evolution of the boundary concentration $C$. We assume that there is a limited amount of surface sites $S$ for species C, so in fact A has to react with a free surface site in order to become $C$ which reflected by the factor $1-u_C$. The same is true for $B$.

\[\begin{aligned} R_{AB}(u_A, u_B)&=k_{AB}^+exp(u_A'(0))u_A - k_{AB}^-exp(-u_A'(0))u_B\\ - D_A \nabla u_A + R_{AB}(u_A, u_B)& =0 \\ - D_B \nabla u_B - R_{AB}(u_A, u_B)& =0 \\ \end{aligned}\]

module Example406_WeirdReaction
using Printf
using VoronoiFVM
using SparseArrays

function main(;n=10,
              Plotter=nothing,
              verbose=false,
              tend=1,
              unknown_storage=:sparse,
              autodetect_sparsity=true)

    h=1.0/convert(Float64,n)
    X=collect(0.0:h:1.0)
    N=length(X)

    grid=VoronoiFVM.Grid(X)
    # By default, \Gamma_1 at X[1] and \Gamma_2 is at X[end]

    # Species numbers
    iA=1
    iB=2
    iC=3


    # Diffusion flux for species A and B
    D_A=1.0
    D_B=1.0e-2
    function flux!(f,u0,edge)
        u=unknowns(edge,u0)
        f[iA]=D_A*(u[iA,1]-u[iA,2])
        f[iB]=D_B*(u[iB,1]-u[iB,2])
    end

    # Storage term of species A and B
    function storage!(f,u,node)
        f[iA]=u[iA]
        f[iB]=u[iB]
    end

    # Source term for species a around 0.5
    function source!(f,node)
        x1=node[1]-0.5
        f[iA]=exp(-100*x1^2)
    end

    # Reaction constants (p = + , m = -)
    # Choosen to prefer path A-> B
    kp_AB=1.0
    km_AB=0.1


    function breaction!(f,u,node)
        if  node.region==1
            R=kp_AB*exp(u[iC])*u[iA] - exp(-u[iC])*km_AB*u[iB]
            f[iA]+=R
            f[iB]-=R
        end
    end

    # This generic operator works on the full solution seen as linear vector, and indexing
    # into it needs to be performed with the help of idx (defined below for a solution vector)
    # Its sparsity is detected automatically using SparsityDetection.jl
    # Here, we calculate the gradient of u_A at the boundary and store the value in u_C which
    # is then used as a parameter in the boundary reaction
    function generic_operator!(f,u,sys)
        f.=0
        f[idx[iC,1]]=u[idx[iC,1]]  + 0.1*(u[idx[iA,1]]-u[idx[iA,2]])/(X[2]-X[1])
    end

If we know the sparsity pattern, we can here create a sparse matrix with values set to 1 in the nonzero slots. This allows to circumvent the autodetection which may takes some time.

    function generic_operator_sparsity(sys)
        idx=unknown_indices(unknowns(sys))
        sparsity=spzeros(num_dof(sys),num_dof(sys))
        sparsity[idx[iC,1],idx[iC,1]]=1
        sparsity[idx[iC,1],idx[iA,1]]=1
        sparsity[idx[iC,1],idx[iA,2]]=1
        sparsity
    end




    if autodetect_sparsity
        physics=VoronoiFVM.Physics(
            num_species=3,
            breaction=breaction!,
            generic=generic_operator!,
            flux=flux!,
            storage=storage!,
            source=source!
        )
    else
        physics=VoronoiFVM.Physics(
            num_species=3,
            breaction=breaction!,
            generic=generic_operator!,
            generic_sparsity=generic_operator_sparsity,
            flux=flux!,
            storage=storage!,
            source=source!
        )
    end

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

    # Enable species in bulk resp
    enable_species!(sys,iA,[1])
    enable_species!(sys,iB,[1])

    # Enable surface species
    enable_boundary_species!(sys,iC,[1])

    # Set Dirichlet bc for species B on \Gamma_2
    boundary_dirichlet!(sys,iB,2,0.0)

    # Initial values
    inival=unknowns(sys)
    inival.=0.0
    U=unknowns(sys)
    idx=unknown_indices(U)

    tstep=0.01
    time=0.0
    T=[]
    u_C=[]

    control=VoronoiFVM.NewtonControl()
    control.verbose=verbose
    while time<tend
        time=time+tstep
        solve!(U,inival,sys,tstep=tstep,control=control)
        inival.=U
        if verbose
            @printf("time=%g\n",time)
        end
        # Record  boundary species
        push!(T,time)
        push!(u_C,U[iC,1])

        if isplots(Plotter)
            Plots=Plotter
            coord=coordinates(grid)
            p1=Plots.plot(coord[1,:],U[iA,:], grid=true, label="[A]")
            Plots.plot!(p1,coord[1,:],U[iB,:], label="[B]",
                        title=@sprintf("max_A=%.5f max_B=%.5f u_C=%.5f\n",maximum(U[iA,:]),maximum(U[iB,:]),u_C[end]),
                        ylabel="[A], [B]", xlabel="x",legend=:topright,framestyle=:full)
            p2=Plots.plot(T,u_C,ylabel="[C]",xlabel="t",framestyle=:full, label="[C]")
            p=Plots.plot(p1,p2,layout=(2,1))
            Plots.gui(p)
        end
    end
    return U[iC,1]
end

function test()
    testval=0.007027597470502758
    main(unknown_storage=:sparse) ≈ testval &&
        main(unknown_storage=:dense) ≈ testval &&
        main(unknown_storage=:sparse,autodetect_sparsity=false) ≈ testval &&
        main(unknown_storage=:dense,autodetect_sparsity=false) ≈ testval
end

end

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