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
using ExtendableGrids
using GridVisualize

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 = simplexgrid(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, u, edge)
        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 = -)
    # Chosen 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(; breaction = breaction!,
                                     generic = generic_operator!,
                                     flux = flux!,
                                     storage = storage!,
                                     source = source!)
    else
        physics = VoronoiFVM.Physics(; 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
    U = unknowns(sys)
    U .= 0.0
    idx = unknown_indices(U)

    tstep = 0.01
    time = 0.0
    T = Float64[]
    u_C = Float64[]

    control = VoronoiFVM.NewtonControl()
    control.verbose = verbose
    p = GridVisualizer(; Plotter = Plotter, layout = (2, 1))
    while time < tend
        time = time + tstep
        U = solve(sys; inival = U, time, tstep, control)
        if verbose
            @printf("time=%g\n", time)
        end
        # Record  boundary pecies
        push!(T, time)
        push!(u_C, U[iC, 1])

        scalarplot!(p[1, 1], grid, U[iA, :]; label = "[A]",
                    title = @sprintf("max_A=%.5f max_B=%.5f u_C=%.5f", maximum(U[iA, :]),
                                     maximum(U[iB, :]), u_C[end]), color = :red)
        scalarplot!(p[1, 1], grid, U[iB, :]; label = "[B]", clear = false, color = :blue)
        scalarplot!(p[2, 1], copy(T), copy(u_C); label = "[C]", clear = true, show = true)
    end
    return U[iC, 1]
end

using Test
function runtests()
    testval = 0.007027597470502758
    @test 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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