Linear Reaction-Diffusion Equations

• Linear Reaction-Diffusion Equations
• • Mathematical Details
  • Implementation
  • Using the Provided Template
  • Just the code

Next, we write a specialised solver for solving linear reaction-diffusion equations. What we produce in this section can also be accessed in FiniteVolumeMethod.LinearReactionDiffusionEquation.

Mathematical Details

To start, let's give the mathematical details. The problems we will be solving take the form

\[\pdv{u}{t} = \div\left[D(\vb x)\grad u\right] + f(\vb x)u.\]

We want to turn this into an equation of the form $\mathrm d\vb u/\mathrm dt = \vb A\vb u + \vb b$ as usual. This takes the same form as our diffusion equation example, except with the extra $f(\vb x)u$ term, which just adds an exta $f(\vb x)$ term to the diagonal of $\vb A$. See the previois sections for further mathematical details.

Implementation

Let us now implement the solver. For constructing $\vb A$, we can use FiniteVolumeMethod.triangle_contributions! as in the previous sections, but we will need an extra function to add $f(\vb x)$ to the appropriate diagonals. We can also reuse apply_dirichlet_conditions!, apply_dudt_conditions, and boundary_edge_contributions! from the diffusion equation example. Here is our implementation.

using FiniteVolumeMethod, SparseArrays, OrdinaryDiffEq, LinearAlgebra
const FVM = FiniteVolumeMethod
function linear_source_contributions!(A, mesh, conditions, source_function, source_parameters)
    for i in each_solid_vertex(mesh.triangulation)
        if !FVM.has_condition(conditions, i)
            x, y = get_point(mesh, i)
            A[i, i] += source_function(x, y, source_parameters)
        end
    end
end
function linear_reaction_diffusion_equation(mesh::FVMGeometry,
    BCs::BoundaryConditions,
    ICs::InternalConditions=InternalConditions();
    diffusion_function,
    diffusion_parameters=nothing,
    source_function,
    source_parameters=nothing,
    initial_condition,
    initial_time=0.0,
    final_time)
    conditions = Conditions(mesh, BCs, ICs)
    n = DelaunayTriangulation.num_solid_vertices(mesh.triangulation)
    Afull = zeros(n + 1, n + 1)
    A = @views Afull[begin:end-1, begin:end-1]
    b = @views Afull[begin:end-1, end]
    _ic = vcat(initial_condition, 1)
    FVM.triangle_contributions!(A, mesh, conditions, diffusion_function, diffusion_parameters)
    FVM.boundary_edge_contributions!(A, b, mesh, conditions, diffusion_function, diffusion_parameters)
    linear_source_contributions!(A, mesh, conditions, source_function, source_parameters)
    FVM.apply_dudt_conditions!(b, mesh, conditions)
    FVM.apply_dirichlet_conditions!(_ic, mesh, conditions)
    FVM.fix_missing_vertices!(A, b, mesh)
    Af = sparse(Afull)
    prob = ODEProblem(MatrixOperator(Af), _ic, (initial_time, final_time))
    return prob
end

linear_reaction_diffusion_equation (generic function with 2 methods)

If you go and look back at the diffusion_equation function from the diffusion equation example, you will see that this is essentially the same function except we now have linear_source_contributions! and source_function and source_parameters arguments.

Let's now test this function. We consider the problem

\[\pdv{T}{t} = \div\left[10^{-3}x^2y\grad T\right] + (x-1)(y-1)T, \quad \vb x \in [0,1]^2,\]

with $\grad T \vdot\vu n = 1$ on the boundary.

using DelaunayTriangulation
tri = triangulate_rectangle(0, 1, 0, 1, 150, 150, single_boundary=true)
mesh = FVMGeometry(tri)
BCs = BoundaryConditions(mesh, (x, y, t, u, p) -> one(x), Neumann)
diffusion_function = (x, y, p) -> p.D * x^2 * y
diffusion_parameters = (D=1e-3,)
source_function = (x, y, p) -> (x - 1) * (y - 1)
initial_condition = [x^2 + y^2 for (x, y) in DelaunayTriangulation.each_point(tri)]
final_time = 8.0
prob = linear_reaction_diffusion_equation(mesh, BCs;
    diffusion_function, diffusion_parameters,
    source_function, initial_condition, final_time)

ODEProblem with uType Vector{Float64} and tType Float64. In-place: true
timespan: (0.0, 8.0)
u0: 22501-element Vector{Float64}:
 0.0
 4.504301608035674e-5
 0.00018017206432142696
 0.00040538714472321063
 0.0007206882572857079
 ⋮
 1.9601369307688843
 1.9733345344804287
 1.986622224224134
 2.0
 1.0

sol = solve(prob, Tsit5(); saveat=2)

retcode: Success
Interpolation: 1st order linear
t: 5-element Vector{Float64}:
 0.0
 2.0
 4.0
 6.0
 8.0
u: 5-element Vector{Vector{Float64}}:
 [0.0, 4.504301608035674e-5, 0.00018017206432142696, 0.00040538714472321063, 0.0007206882572857079, 0.0011260754020089186, 0.0016215485788928425, 0.0022071077879374803, 0.0028827530291428314, 0.0036484843025088956  …  1.8955002026935723, 1.9082473762443133, 1.921084635827215, 1.9340119814422772, 1.9470294130895003, 1.9601369307688843, 1.9733345344804287, 1.986622224224134, 2.0, 1.0]
 [5.392972715241143e-10, 0.0003283969313676684, 0.0012960759990657792, 0.002877290944573626, 0.005046982038098889, 0.007780761845897448, 0.011054900287088444, 0.014846309987847438, 0.019132531927487165, 0.023891721371031467  …  1.8576012589880762, 1.8669187520861235, 1.8756974896164818, 1.884763887437326, 1.8917905879259957, 1.902881675506313, 1.9023588034293868, 1.9299909658674865, 1.8825037936504285, 1.0]
 [7.916735417539211e-9, 0.002394255277378063, 0.009323370071034556, 0.02042192008254055, 0.03534389913584281, 0.05376187635010653, 0.07536617230272592, 0.09986406790939065, 0.1269790448138673, 0.15645005612239932  …  1.8509136105071604, 1.858591600062336, 1.8664895899891367, 1.8731272794777616, 1.8825835384209586, 1.884116496741108, 1.905311415327742, 1.8762173442191878, 1.978490952695312, 1.0]
 [8.716291359760346e-8, 0.017455872288248277, 0.06706791816304723, 0.14494671157302366, 0.24751160642335782, 0.37147055211194413, 0.5138009456452572, 0.6717316202228347, 0.8427259076096069, 1.0244657148949718  …  1.856052705661518, 1.8629729344642054, 1.869629885146697, 1.8768276952756373, 1.8824053751208059, 1.892111089683071, 1.8912408569476635, 1.9171352613990147, 1.8728687859751985, 1.0]
 [8.530436664276854e-7, 0.1272660020066086, 0.48245430288397767, 1.0287720968254588, 1.7333054836766988, 2.5666820729791873, 3.5027585194801247, 4.5183326883131265, 5.592878646428236, 6.708302802132749  …  1.870052628652889, 1.8761847240187424, 1.8825065062351822, 1.8884430769360512, 1.8956204930445033, 1.8997921589345663, 1.9118930626122232, 1.9042189119217956, 1.9487418307409279, 1.0]

using CairoMakie
fig = Figure(fontsize=38)
for j in eachindex(sol)
    ax = Axis(fig[1, j], width=600, height=600,
        xlabel="x", ylabel="y",
        title="t = $(sol.t[j])")
    tricontourf!(ax, tri, sol.u[j], levels=0:0.1:1, extendlow=:auto, extendhigh=:auto, colormap=:turbo)
    tightlimits!(ax)
end
resize_to_layout!(fig)
fig

Here is how we could convert this into an FVMProblem. Note that the Neumann boundary conditions are expressed as $\grad T\vdot\vu n = 1$ above, but for FVMProblem we need them in the form $\vb q\vdot\vu n = \ldots$. For this problem, $\vb q=-D\grad T$, which gives $\vb q\vdot\vu n = -D$.

_BCs = BoundaryConditions(mesh, (x, y, t, u, p) -> -p.D(x, y, p.Dp), Neumann;
    parameters=(D=diffusion_function, Dp=diffusion_parameters))
fvm_prob = FVMProblem(
    mesh,
    _BCs;
    diffusion_function=let D=diffusion_function
        (x, y, t, u, p) -> D(x, y, p)
    end,
    diffusion_parameters=diffusion_parameters,
    source_function=let S=source_function
        (x, y, t, u, p) -> S(x, y, p) * u
    end,
    final_time=final_time,
    initial_condition=initial_condition
)
fvm_sol = solve(fvm_prob, Tsit5(), saveat=2.0)

retcode: Success
Interpolation: 1st order linear
t: 5-element Vector{Float64}:
 0.0
 2.0
 4.0
 6.0
 8.0
u: 5-element Vector{Vector{Float64}}:
 [0.0, 4.504301608035674e-5, 0.00018017206432142696, 0.00040538714472321063, 0.0007206882572857079, 0.0011260754020089186, 0.0016215485788928425, 0.0022071077879374803, 0.0028827530291428314, 0.0036484843025088956  …  1.882843115174992, 1.8955002026935723, 1.9082473762443133, 1.921084635827215, 1.9340119814422772, 1.9470294130895003, 1.9601369307688843, 1.9733345344804287, 1.986622224224134, 2.0]
 [5.392972715241146e-10, 0.00032839693136766865, 0.0012960759990657787, 0.002877290944573627, 0.0050469820380988906, 0.00778076184589744, 0.01105490028708845, 0.014846309987847429, 0.01913253192748716, 0.02389172137103148  …  1.8480635912090653, 1.8576012582625914, 1.8669187542720502, 1.8756974833133426, 1.8847639048460716, 1.8917905417452556, 1.902881793954016, 1.9023585053824088, 1.9299917234623845, 1.8825017475754324]
 [7.916735417539227e-9, 0.0023942552773780694, 0.009323370071034566, 0.020421920082540564, 0.03534389913584279, 0.0537618763501065, 0.07536617230272606, 0.09986406790939072, 0.12697904481386746, 0.15645005612239948  …  1.8429322427231227, 1.8509136099818753, 1.8585916016450483, 1.866489585425376, 1.8731272920825013, 1.8825835049839645, 1.884116582502732, 1.9053111995279264, 1.8762178927529864, 1.9784894712422374]
 [8.716291359760358e-8, 0.017455872288248315, 0.06706791816304726, 0.14494671157302375, 0.24751160642335737, 0.37147055211194385, 0.5138009456452574, 0.671731620222835, 0.8427259076096064, 1.0244657148949723  …  1.8491360538216242, 1.8560527070463413, 1.8629729302916793, 1.8696298971782281, 1.8768276620455648, 1.8824054632712968, 1.8921108635883268, 1.8912414258641945, 1.9171338152905444, 1.8728726915537697]
 [8.530436664276897e-7, 0.12726600200660945, 0.4824543028839793, 1.0287720968254626, 1.7333054836767012, 2.566682072979191, 3.5027585194801385, 4.518332688313142, 5.59287864642825, 6.708302802132766  …  1.8638903357449816, 1.8700526280569478, 1.8761847258143487, 1.8825065010575315, 1.8884430912362884, 1.8956204551098073, 1.8997922562322205, 1.9118928177845889, 1.9042195342403745, 1.9487401500135615]

Using the Provided Template

The above code is implemented in LinearReactionDiffusionEquation in FiniteVolumeMethod.jl.

prob = LinearReactionDiffusionEquation(mesh, BCs;
    diffusion_function, diffusion_parameters,
    source_function,  initial_condition, final_time)
sol = solve(prob, Tsit5(); saveat=2)

retcode: Success
Interpolation: 1st order linear
t: 5-element Vector{Float64}:
 0.0
 2.0
 4.0
 6.0
 8.0
u: 5-element Vector{Vector{Float64}}:
 [0.0, 4.504301608035674e-5, 0.00018017206432142696, 0.00040538714472321063, 0.0007206882572857079, 0.0011260754020089186, 0.0016215485788928425, 0.0022071077879374803, 0.0028827530291428314, 0.0036484843025088956  …  1.8955002026935723, 1.9082473762443133, 1.921084635827215, 1.9340119814422772, 1.9470294130895003, 1.9601369307688843, 1.9733345344804287, 1.986622224224134, 2.0, 1.0]
 [5.392972715241143e-10, 0.0003283969313676684, 0.0012960759990657792, 0.002877290944573626, 0.005046982038098889, 0.007780761845897448, 0.011054900287088444, 0.014846309987847438, 0.019132531927487165, 0.023891721371031467  …  1.8576012589880762, 1.8669187520861235, 1.8756974896164818, 1.884763887437326, 1.8917905879259957, 1.902881675506313, 1.9023588034293868, 1.9299909658674865, 1.8825037936504285, 1.0]
 [7.916735417539211e-9, 0.002394255277378063, 0.009323370071034556, 0.02042192008254055, 0.03534389913584281, 0.05376187635010653, 0.07536617230272592, 0.09986406790939065, 0.1269790448138673, 0.15645005612239932  …  1.8509136105071604, 1.858591600062336, 1.8664895899891367, 1.8731272794777616, 1.8825835384209586, 1.884116496741108, 1.905311415327742, 1.8762173442191878, 1.978490952695312, 1.0]
 [8.716291359760346e-8, 0.017455872288248277, 0.06706791816304723, 0.14494671157302366, 0.24751160642335782, 0.37147055211194413, 0.5138009456452572, 0.6717316202228347, 0.8427259076096069, 1.0244657148949718  …  1.856052705661518, 1.8629729344642054, 1.869629885146697, 1.8768276952756373, 1.8824053751208059, 1.892111089683071, 1.8912408569476635, 1.9171352613990147, 1.8728687859751985, 1.0]
 [8.530436664276854e-7, 0.1272660020066086, 0.48245430288397767, 1.0287720968254588, 1.7333054836766988, 2.5666820729791873, 3.5027585194801247, 4.5183326883131265, 5.592878646428236, 6.708302802132749  …  1.870052628652889, 1.8761847240187424, 1.8825065062351822, 1.8884430769360512, 1.8956204930445033, 1.8997921589345663, 1.9118930626122232, 1.9042189119217956, 1.9487418307409279, 1.0]

Here is a benchmark comparison of LinearReactionDiffusionEquation versus FVMProblem.

using BenchmarkTools
using Sundials
@btime solve($prob, $CVODE_BDF(linear_solver=:GMRES); saveat=$2);

  48.360 ms (1087 allocations: 1.58 MiB)

@btime solve($fvm_prob, $CVODE_BDF(linear_solver=:GMRES); saveat=$2);

  163.686 ms (83267 allocations: 90.84 MiB)

Just the code

An uncommented version of this example is given below. You can view the source code for this file here.

using FiniteVolumeMethod, SparseArrays, OrdinaryDiffEq, LinearAlgebra
const FVM = FiniteVolumeMethod
function linear_source_contributions!(A, mesh, conditions, source_function, source_parameters)
    for i in each_solid_vertex(mesh.triangulation)
        if !FVM.has_condition(conditions, i)
            x, y = get_point(mesh, i)
            A[i, i] += source_function(x, y, source_parameters)
        end
    end
end
function linear_reaction_diffusion_equation(mesh::FVMGeometry,
    BCs::BoundaryConditions,
    ICs::InternalConditions=InternalConditions();
    diffusion_function,
    diffusion_parameters=nothing,
    source_function,
    source_parameters=nothing,
    initial_condition,
    initial_time=0.0,
    final_time)
    conditions = Conditions(mesh, BCs, ICs)
    n = DelaunayTriangulation.num_solid_vertices(mesh.triangulation)
    Afull = zeros(n + 1, n + 1)
    A = @views Afull[begin:end-1, begin:end-1]
    b = @views Afull[begin:end-1, end]
    _ic = vcat(initial_condition, 1)
    FVM.triangle_contributions!(A, mesh, conditions, diffusion_function, diffusion_parameters)
    FVM.boundary_edge_contributions!(A, b, mesh, conditions, diffusion_function, diffusion_parameters)
    linear_source_contributions!(A, mesh, conditions, source_function, source_parameters)
    FVM.apply_dudt_conditions!(b, mesh, conditions)
    FVM.apply_dirichlet_conditions!(_ic, mesh, conditions)
    FVM.fix_missing_vertices!(A, b, mesh)
    Af = sparse(Afull)
    prob = ODEProblem(MatrixOperator(Af), _ic, (initial_time, final_time))
    return prob
end

using DelaunayTriangulation
tri = triangulate_rectangle(0, 1, 0, 1, 150, 150, single_boundary=true)
mesh = FVMGeometry(tri)
BCs = BoundaryConditions(mesh, (x, y, t, u, p) -> one(x), Neumann)
diffusion_function = (x, y, p) -> p.D * x^2 * y
diffusion_parameters = (D=1e-3,)
source_function = (x, y, p) -> (x - 1) * (y - 1)
initial_condition = [x^2 + y^2 for (x, y) in DelaunayTriangulation.each_point(tri)]
final_time = 8.0
prob = linear_reaction_diffusion_equation(mesh, BCs;
    diffusion_function, diffusion_parameters,
    source_function, initial_condition, final_time)

sol = solve(prob, Tsit5(); saveat=2)

using CairoMakie
fig = Figure(fontsize=38)
for j in eachindex(sol)
    ax = Axis(fig[1, j], width=600, height=600,
        xlabel="x", ylabel="y",
        title="t = $(sol.t[j])")
    tricontourf!(ax, tri, sol.u[j], levels=0:0.1:1, extendlow=:auto, extendhigh=:auto, colormap=:turbo)
    tightlimits!(ax)
end
resize_to_layout!(fig)
fig

_BCs = BoundaryConditions(mesh, (x, y, t, u, p) -> -p.D(x, y, p.Dp), Neumann;
    parameters=(D=diffusion_function, Dp=diffusion_parameters))
fvm_prob = FVMProblem(
    mesh,
    _BCs;
    diffusion_function=let D=diffusion_function
        (x, y, t, u, p) -> D(x, y, p)
    end,
    diffusion_parameters=diffusion_parameters,
    source_function=let S=source_function
        (x, y, t, u, p) -> S(x, y, p) * u
    end,
    final_time=final_time,
    initial_condition=initial_condition
)
fvm_sol = solve(fvm_prob, Tsit5(), saveat=2.0)

prob = LinearReactionDiffusionEquation(mesh, BCs;
    diffusion_function, diffusion_parameters,
    source_function,  initial_condition, final_time)
sol = solve(prob, Tsit5(); saveat=2)

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