105: 1D Nonlinear Poisson equation

(source code)

Solve the nonlinear Poisson equation

in $\Omega=(0,1)$ with boundary condition $u(0)=0$ and $u(1)=1$ with

This stationary problem is an example of a nonlinear Poisson equation or Poisson-Boltzmann equation. Such equation occur e.g. in simulations of electrochemical systems and semicondutor devices.

module Example105_NonlinearPoisson1D
using Printf
using VoronoiFVM

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

    # Create a one-dimensional discretization
    h=1.0/convert(Float64,n)
    grid=VoronoiFVM.Grid(collect(0:h:1))

    # A parameter which is "passed" to the flux function via scope
    ϵ=1.0e-3

    # Flux function which describes the flux
    # between neigboring control volumes
    function flux!(f,u,edge)
        f[1]=ϵ*(u[1,1]-u[1,2])
    end

    # Source term
    function source!(f,node)
        if node[1]<=0.5
            f[1]=1
        else
            f[1]=-1
        end
    end

    # Reaction term
    function reaction!(f,u,node)
        f[1]=exp(u[1]) - exp(-u[1])
    end

    # Create a physics structure
    physics=VoronoiFVM.Physics(
        flux=flux!,
        source=source!,
        reaction=reaction!)


    # Create a finite volume system - either
    # in the dense or  the sparse version.
    # The difference is in the way the solution object
    # is stored - as dense or as sparse matrix

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

    # Add species 1 to region 1
    enable_species!(sys,1,[1])

    # Set boundary conditions
    boundary_dirichlet!(sys,1,1,0.0)
    boundary_dirichlet!(sys,1,2,1.0)

    # Create a solution array
    inival=unknowns(sys,inival=0.5)
    solution=unknowns(sys)

    # Create solver control info
    control=VoronoiFVM.NewtonControl()
    control.verbose=verbose

    # Stationary solution of the problem
    solve!(solution,inival,sys, control=control)

    if isplots(Plotter)
        Plotter.plot(grid.coord[1,:],solution[1,:],
                   label="",
                   title="Nonlinear Poisson",
                   grid=true,show=true)
    end

    return sum(solution)
end

function test()
    testval=1.5247901344230088
    main(unknown_storage=:sparse) ≈ testval && main(unknown_storage=:dense) ≈ testval
end

end

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