VoronoiFVM.jl: Tipps and Examples

5.0 μs

Grid generation

VoronoiFVM works on simplicial grids provided by the package ExtendableGrids.jl

There are several ways to create a grid.

8.1 μs

1D grids

1D grids are created from a vector of monotonicaly increasing x-axis positions.

6.0 μs

Float641

0.0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1.0

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# Create a 1D vector:
X=collect(range(0,1,length=21))

104 ms

grid1d_a

ExtendableGrids.ExtendableGrid{Float64, Int32};
dim: 1 nodes: 21 cells: 20 bfaces: 2

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# Create grid from vector:
grid1d_a=ExtendableGrids.simplexgrid(X)

247 ms

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# Visualize grid
GridVisualize.gridplot(grid1d_a,resolution=(600,200),Plotter=PyPlot)

2.6 s

As we see, the grid is chacracterized by interior points, and boundary points. Each grid cell is endowed with a region number (for allowing different physics, parameters etc. for different regions). Each boundary node has a boundary region number, which is meant to be used to distinguish different boundary conditions.

More sophisticated grids can be created, as we see in the following example:

31.3 μs

grid1d_b

ExtendableGrids.ExtendableGrid{Float64, Int32};
dim: 1 nodes: 53 cells: 52 bfaces: 3

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grid1d_b=let
    hmax=0.1
    hmin=0.01
    # Create vectors with geometric distributions of interval sizes
    X1=ExtendableGrids.geomspace(0.0,1.0,hmax,hmin)
    X2=geomspace(1.0,2.0,hmin,hmax)
    # Glue them together at common point x=1 (this is different from vcat!)
    X3=glue(X1,X2)
    grid1d_b=simplexgrid(X3)
    # Mark an additional interior boundary point at x=1
    ExtendableGrids.bfacemask!(grid1d_b,[1.0],[1.0],3)
    # Change cell region number at the right part
    ExtendableGrids.cellmask!(grid1d_b,[1.0],[2.0],2)
    grid1d_b
end

527 ms

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gridplot(grid1d_b,resolution=(600,200),Plotter=PyPlot,aspect=0.5)

236 ms

2D Tensor product grids

These are created from two vectors of x and y coordinates, respectively. This results in the creation of a grid of quadrilaterals. Then, each of them is subdivided into two triangles, resulting in a boundary conforming Delaunay grid.

5.3 μs

grid2d_a

ExtendableGrids.ExtendableGrid{Float64, Int32};
dim: 2 nodes: 121 cells: 200 bfaces: 40

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grid2d_a=let
    X=collect(range(0,1,length=11))
    Y=collect(range(0,1,length=11))
    simplexgrid(X,Y)
end

136 ms

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gridplot(grid2d_a,resolution=(600,200),Plotter=PyPlot,legend_location=(1.5,0))

1.7 s

Once again, we see a default distrbution of cell regions and boundary regions. This can be modified in a similar manner as in the 1D case.

2.4 μs

grid2d_b

ExtendableGrids.ExtendableGrid{Float64, Int32};
dim: 2 nodes: 121 cells: 200 bfaces: 56, edges: 320

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grid2d_b=let
    X=collect(range(0,1,length=11))
    Y=collect(range(0,1,length=11))
    grid=simplexgrid(X,Y)
    cellmask!(grid,[0.3,0.3],[0.7,0.7],2)
    bfacemask!(grid,[0.3,0.3],[0.3,0.7],5)  
    bfacemask!(grid,[0.3,0.7],[0.7,0.7],6)  
    bfacemask!(grid,[0.7,0.3],[0.7,0.7],7)  
    bfacemask!(grid,[0.3,0.3],[0.7,0.3],8)  
    grid
end

905 ms

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gridplot(grid2d_b,resolution=(600,200),Plotter=PyPlot,legend_location=(1.5,0))

117 ms

2D Unstructured grids

These can be created using the mesh generator Triangle (by J. Shewchuk) via the packages Triangulate.jl and SimplexGridFactory.jl.

5.4 μs

builder2d (generic function with 1 method)

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function builder2d()
    b=SimplexGridFactory.SimplexGridBuilder(Generator=Triangulate)
    p1=point!(b,0,0)
    p2=point!(b,0,1)
    p3=point!(b,0.5,0)
    facetregion!(b,1)
    facet!(b,p1,p2)
    facetregion!(b,2)
    facet!(b,p2,p3)
    facetregion!(b,3)
    facet!(b,p1,p3) 
    point!(b,0.1,0.1)
    b
end

59.6 μs

builder

SimplexGridBuilderGenerator

Triangulate

current_facetregion

current_cellregion

current_cellvolume

1.0

point_identity_tolerance

1.0e-12

facetregionsInt321

facetsVector{Int32}1Int321

2Int321

3Int321

pointlistBinnedPointList{Float64}dim

tol

1.0e-12

binning_region_minFloat641

1.79769e308

1.79769e308

binning_region_maxFloat641

-1.79769e308

-1.79769e308

binning_region_increase_factor

0.01

points

2×4 ElasticArrays.ElasticMatrix{Float64, 1, Vector{Float64}}:
 0.0  0.0  0.5  0.1
 0.0  1.0  0.0  0.1

bins

0×0 Matrix{Vector{Int32}}

number_of_directional_bins

unbinnedInt321

num_allowed_unbinned_points

max_unbinned_ratio

0.05

current_binInt321

regionpoints

2×0 ElasticArrays.ElasticMatrix{Float64, 1, Vector{Float64}}

regionnumbersInt32regionvolumesFloat64optionsDict

:optlevel

:attributes

true

:quiet

true

:verbose

false

:addflags

""

:maxvolume

Inf

:confdelaunay

true

:unsuitable

nothing

:debugfacets

true

:volumecontrol

true

:quality

true

:flags

nothing

:refine

false

:PLC

true

:nosteiner

false

:minangle

:check

false

checkexisting

true

345 ms

For debugging purposes, the current state of the builder and its possible output can be visualized:

2.1 μs

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builderplot(builder,Plotter=PyPlot)

1.4 s

Finally, we can create a grid from the builder:

2.0 μs

grid2d_c

ExtendableGrids.ExtendableGrid{Float64, Int32};
dim: 2 nodes: 230 cells: 381 bfaces: 77

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grid2d_c=simplexgrid(builder,maxvolume=0.001)

335 ms

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gridplot(grid2d_c,resolution=(600,200),Plotter=PyPlot,legend_location=(2,0))

206 ms

Stationary scalar problems

2.4 μs

Diffusion with Dirichlet boundary conditions

This is mathematically similar to heat conduction and other problems.

$\begin{aligned} - \nabla \cdot D \nabla u & = 10 \\ u_{Γ_{e a s t}} & = 0 \\ u_{Γ_{w e s t}} & = 1 \\ D \nabla u \cdot \vec{n} |_{\partial Ω ∖ (Γ_{e a s t} \cup Γ_{w e s t})} & = 0 \end{aligned}$

Besides of the domain and its boundary it is characterize by a flux term and a source term.

3.4 μs

solve_diffproblem_dirichlet (generic function with 1 method)

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function solve_diffproblem_dirichlet(grid;D=1.0)
    species1=1
    
    # Use finite difference flux between disretization points. 
    # Division by distance and multiplication by interface size
    # is done by the VoronoiFVM Module.
    function flux(f,u0,edge)
        u=unknowns(edge,u0)
        f[species1]=D*(u[species1,1]-u[species1,2])
    end
    
    # Specify a constant source term
    function source(f,node)
        f[species1]=10
    end
​
    # Combine flux and source to "physics"
    physics=VoronoiFVM.Physics(flux=flux,source=source)
    
    # Create system from physics and grid
    system=VoronoiFVM.System(grid,physics)
​
    # Enable species in cellregion 1
    enable_species!(system,species1,[1])
​
    
    # Enable boundary conditions. For those boundary regions
    # which are not specified here, by default, homogeneous 
    # Neumann boundary conditions are assumed.
    west=dim_space(grid)==1  ? 1 : 4
    east=2
    boundary_dirichlet!(system, species1, west, 0)
    boundary_dirichlet!(system, species1, east, 1)
​
    # Solve with given initial value
    solve(unknowns(system,inival=0),system)
end

69.9 μs

solution1d_a

1×21 Matrix{Float64}:
 6.0e-30  0.2875  0.55  0.7875  1.0  …  1.6875  1.6  1.4875  1.35  1.1875  1.0

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solution1d_a=solve_diffproblem_dirichlet(grid1d_a)

1.5 s

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scalarplot(grid1d_a,solution1d_a[1,:],Plotter=PyPlot)

397 ms

solution2d_a

1×121 Matrix{Float64}:
 3.0e-31  0.55  1.0  1.35  1.6  1.75  1.8  …  1.6  1.75  1.8  1.75  1.6  1.35  1.0

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solution2d_a=solve_diffproblem_dirichlet(grid2d_a)

33.3 ms

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scalarplot(grid2d_a,solution2d_a[1,:],Plotter=PyPlot)

516 ms

Diffusion with Robin boundary conditions

$\begin{aligned} - \nabla \cdot D \nabla u & = 10 \\ D \nabla u \cdot \vec{n} + a u & = 0 on Γ_{e a s t} \\ D \nabla u \cdot \vec{n} + a u & = a on Γ_{e a s t} \\ D \nabla u \cdot \vec{n} |_{\partial Ω ∖ (Γ_{e a s t} \cup Γ_{w e s t})} & = 0 \end{aligned}$

4.1 μs

solve_diffproblem_robin (generic function with 1 method)

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function solve_diffproblem_robin(grid;D=1.0,a=0.5)
    species1=1
​
    function flux(f,u0,edge)
        u=unknowns(edge,u0)
        f[species1]=D*(u[species1,1]-u[species1,2])
    end
​
    function source(f,node)
        f[species1]=10
    end
​
    physics=VoronoiFVM.Physics(flux=flux,source=source)
​
    system=VoronoiFVM.System(grid,physics)
​
    enable_species!(system,species1,[1])
​
    west=dim_space(grid)==1  ? 1 : 4
    east=2
    boundary_robin!(system, species1, west, a, 0)
    boundary_robin!(system, species1, east, a, a*1)
​
    solve(unknowns(system,inival=0),system)
end

76.4 μs

solution1d_robin

1×21 Matrix{Float64}:
 5.33333  5.5875  5.81667  6.02083  6.2  …  6.4  6.25417  6.08333  5.8875  5.66667

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solution1d_robin=solve_diffproblem_robin(grid1d_a,a=1)

880 ms

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scalarplot(grid1d_a,solution1d_robin[1,:],Plotter=PyPlot)

60.6 ms

solution2d_robin

1×121 Matrix{Float64}:
 5.33333  5.81667  6.2  6.48333  6.66667  …  6.73333  6.61667  6.4  6.08333  5.66667

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solution2d_robin=solve_diffproblem_robin(grid2d_a,a=1)

2.4 ms

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scalarplot(grid2d_a,solution2d_robin[1,:],Plotter=PyPlot)

74.3 ms

Stationary Reaction-Diffusion problem

Here, we regard two species $u_{1}, u_{2}$ , and a reaction converting $u_{1}$ into $u_{2}$ . Dirichlet boundary conditions "inject" $u_{1}$ an "remove" $u_{2}$ .

$\begin{aligned} - \nabla \cdot D_{1} \nabla u_{1} + r (u_{1}) & = 0 \\ - \nabla \cdot D_{2} \nabla u_{2} - r (u_{1}) & = 0 \\ r (u_{1}) & = k u_{1} \\ u_{1} |_{Γ_{w e s t}} & = 1 \\ u_{2} |_{Γ_{e a s t}} & = 0 \end{aligned}$

Boundary conditons not specified are assumed to be homogeneous Neumann.

4.9 μs

solve_readiff (generic function with 1 method)

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function solve_readiff(grid;D_1=1.0,D_2=1.0,k=1)
    species1=1
    species2=2
​
    function flux(f,u0,edge)
        u=unknowns(edge,u0)
        f[species1]=D_1*(u[species1,1]-u[species1,2])
        f[species2]=D_2*(u[species2,1]-u[species2,2])
    end
​
    function reaction(f,u0,node)
        u=unknowns(node,u0)
        r=k*u[species1]
        f[species1]=r
        f[species2]=-r
    end
​
    physics=VoronoiFVM.Physics(num_species=2,flux=flux,reaction=reaction)
​
    system=VoronoiFVM.System(grid,physics)
​
    enable_species!(system,species1,[1])
    enable_species!(system,species2,[1])
​
    west=dim_space(grid)==1  ? 1 : 4
    east=2
    boundary_dirichlet!(system, species1, west,1)
    boundary_dirichlet!(system, species2, east,0)
​
    solve(unknowns(system,inival=0),system)
end
​

87.7 μs

solution_readiff_1d

2×21 Matrix{Float64}:
 1.0      0.854464  0.730289  0.624372  …  0.0890498  0.0858439  0.0847841
 2.24551  2.23301   2.19915   2.14703      0.311807   0.156976   3.16072e-30

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solution_readiff_1d=solve_readiff(grid1d_a,k=10, D_2=1)

1.3 s

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let
    v=GridVisualizer(Plotter=PyPlot)
    scalarplot!(v[1,1],grid1d_a, solution_readiff_1d[1,:],label="spec1", color=:red)
    scalarplot!(v[1,1],grid1d_a, solution_readiff_1d[2,:],label="spec2", color=:green,show=true,clear=false)
end

264 ms

solution_readiff_2d

2×121 Matrix{Float64}:
 1.0      0.884085  0.785851  0.703335  0.634885  …  0.478053  0.464174  0.459578
 0.71873  0.70873   0.681048  0.63765   0.580184     0.233355  0.121319  6.29576e-32

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solution_readiff_2d=solve_readiff(grid2d_a,k=2)

1.2 s

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let
    v=GridVisualizer(Plotter=PyPlot,layout=(1,2))
    scalarplot!(v[1,1],grid2d_a, solution_readiff_2d[1,:],label="spec1",flimits=(0,1))
    scalarplot!(v[1,2],grid2d_a, solution_readiff_2d[2,:],label="spec2",flimits=(0,1), show=true)
end

228 ms

Transient Reaction-Diffusion problem

Here, we regard two species $u_{1}, u_{2}$ , and a reaction converting $u_{1}$ into $u_{2}$ . Dirichlet boundary conditions "inject" $u_{1}$ an "remove" $u_{2}$ .

$\begin{aligned} \partial_{t} u_{1} - \nabla \cdot D_{1} \nabla u_{1} + r (u_{1}) & = 0 \\ \partial_{t} u_{2} - \nabla \cdot D_{2} \nabla u_{2} - r (u_{1}) & = 0 \\ r (u_{1}) & = k u_{1} \\ u_{1} |_{Γ_{w e s t}} & = 1 \\ u_{2} |_{Γ_{e a s t}} & = 0 \\ u_{1} |_{t = 0} & = 0 \\ u_{2} |_{t = 0} & = 0 \end{aligned}$

Boundary conditons not specified are assumed to be homogeneous Neumann.

6.6 μs

evolution (generic function with 1 method)

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# Function describing evolution of system with initial value inival 
# using the Implicit Euler method
function evolution(inival, # initial value
                   sys,    # finite volume system
                   grid,   # simplex grid
                   tstep,  # initial time step 
                   tend,   # end time 
                   dtgrowth  # time step growth factor
                   )
    time=0.0
    # record time and solution
    times=[time]
    solutions=[copy(inival)]
    
    solution=copy(inival)
    while time<tend
        time=time+tstep
        solve!(solution,inival,sys,tstep=tstep) # solve implicit Euler time step 
        inival.=solution  # copy solution to inivalue
        push!(times,time) 
        push!(solutions,copy(solution))
        tstep*=dtgrowth  # increase timestep by factor when approaching stationary state
    end
    # return result and grid 
    (times=times,solutions=solutions,grid=grid)
end
​

32.4 μs

transient_reaction_diffusion (generic function with 1 method)

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function transient_reaction_diffusion(grid;D_1=1.0,D_2=1.0,k=1,
    tstep=1.0e-3,tend=1,dtgrowth=1.1)
    species1=1
    species2=2
​
    function flux(f,u0,edge)
        u=unknowns(edge,u0)
        f[species1]=D_1*(u[species1,1]-u[species1,2])
        f[species2]=D_2*(u[species2,1]-u[species2,2])
    end
​
    function reaction(f,u0,node)
        u=unknowns(node,u0)
        r=k*u[species1]
        f[species1]=r
        f[species2]=-r
    end
​
    function storage(f,u,node)
        f.=u
    end
​
    physics=VoronoiFVM.Physics(num_species=2,flux=flux,reaction=reaction,storage=storage)
​
    system=VoronoiFVM.System(grid,physics)
​
    enable_species!(system,species1,[1])
    enable_species!(system,species2,[1])
​
    west=dim_space(grid)==1  ? 1 : 4
    east=2
    boundary_dirichlet!(system, species1, west,1)
    boundary_dirichlet!(system, species2, east,0)
    ## Create a solution array
    inival=unknowns(system,inival=0)
    evolution(inival,system,grid,tstep,tend,dtgrowth)   
end

158 μs

tsol_readiff

timesFloat641

0.0

0.001

0.0021

0.00331

0.004641

0.0061051

0.00771561

0.00948717

0.0114359

0.0135795

0.0159374

0.0185312

0.0213843

0.0245227

0.027975

0.0317725

0.0359497

0.0405447

0.0455992

0.0511591

more89

43.8993

48.2902

53.1202

58.4332

64.2776

70.7063

77.778

85.5568

94.1134

103.526

solutionsMatrix{Float64}1

2×21 Matrix{Float64}:
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0

2×21 Matrix{Float64}:
 1.0         0.23429      0.0548919   …  4.53819e-12  1.11825e-12  4.96724e-13
 0.00069226  0.000307586  0.00010615     5.37779e-14  1.25719e-14  2.63856e-43

2×21 Matrix{Float64}:
 1.0         0.388729     0.129427     …  9.70382e-11  2.62235e-11  1.25317e-11
 0.00131875  0.000780668  0.000345951     1.26334e-12  3.17705e-13  6.6674e-42

2×21 Matrix{Float64}:
 1.0         0.493864    0.206185     …  1.06107e-9   3.13559e-10  1.60499e-10
 0.00195214  0.00135646  0.000715781     1.51357e-11  4.07663e-12  8.55451e-41

2×21 Matrix{Float64}:
 1.0         0.568244   0.277615    …  7.89166e-9   2.543e-9     1.38824e-9
 0.00262716  0.0020111  0.00120404     1.23024e-10  3.53349e-11  7.41403e-40

2×21 Matrix{Float64}:
 1.0         0.623046    0.341224    …  4.48229e-8  1.57063e-8   9.10588e-9
 0.00336164  0.00273872  0.00180061     7.6171e-10  2.32314e-10  4.87393e-39

2×21 Matrix{Float64}:
 1.0         0.665021   0.396862    …  2.0705e-7   7.8674e-8   4.82398e-8
 0.00416662  0.0035414  0.00249964     3.82598e-9  1.23393e-9  2.58846e-38

2×21 Matrix{Float64}:
 1.0         0.6983      0.445307   …  8.09271e-7  3.32497e-7  2.14744e-7
 0.00505087  0.00442479  0.0032996     1.62192e-8  5.50887e-9  1.15546e-37

2×21 Matrix{Float64}:
 1.0         0.725463   0.487589    0.305176    …  2.75049e-6  1.21828e-6  8.25489e-7
 0.00602284  0.0053963  0.00420244  0.00294787     5.96309e-8  2.12446e-8  4.4553e-37

2×21 Matrix{Float64}:
 1.0         0.748172    0.5247      …  8.29339e-6  3.94783e-6  2.79554e-6
 0.00709136  0.00646445  0.00521286     1.93965e-7  7.22023e-8  1.51393e-36

2×21 Matrix{Float64}:
 1.0         0.767529    0.557499    …  2.25305e-5  1.14884e-5  8.46976e-6
 0.00826603  0.00763874  0.00633768     5.66815e-7  2.19619e-7  4.60412e-36

2×21 Matrix{Float64}:
 1.0         0.784289    0.586693    …  5.58339e-5  3.03906e-5  2.32422e-5
 0.00955732  0.00892964  0.00758547     1.50636e-6  6.05297e-7  1.2687e-35

2×21 Matrix{Float64}:
 1.0        0.798987   0.612852    …  0.000127507  7.38152e-5  5.83576e-5
 0.0109767  0.0103486  0.00896638     3.67743e-6   1.52712e-6  3.20013e-35

2×21 Matrix{Float64}:
 1.0        0.812014   0.636436  0.481235    …  0.000270654  0.000166012  0.000135227
 0.0125369  0.0119083  0.010492  0.00869996     8.31701e-6   3.55739e-6   7.45284e-35

2×21 Matrix{Float64}:
 1.0        0.823661   0.657814   0.509094  …  0.000537948  0.000348231  0.000291341
 0.0142516  0.0136224  0.0121755  0.010303     1.75533e-5   7.70879e-6   1.61459e-34

2×21 Matrix{Float64}:
 1.0       0.834148   0.677286   0.534937   …  0.00100763  0.000685612  0.000587419
 0.016136  0.0155062  0.0140313  0.0120849     3.47922e-5  1.56418e-5   3.27522e-34

2×21 Matrix{Float64}:
 1.0        0.84365    0.695096   0.558939   …  0.00178873  0.00127411  0.00111488
 0.0182066  0.0175763  0.0160757  0.0140608     6.5131e-5   2.98935e-5  6.25741e-34

2×21 Matrix{Float64}:
 1.0        0.852303   0.711445   0.581259   …  0.00302446   0.00224611  0.00200222
 0.0204818  0.0198507  0.0183264  0.0162479     0.000115737  5.40928e-5  1.13191e-33

2×21 Matrix{Float64}:
 1.0        0.860217   0.726498  0.602042   …  0.00489262   0.00377326  0.00341869
 0.0229814  0.0223495  0.020803  0.0186653     0.000196115  9.3122e-5   1.94791e-33

2×21 Matrix{Float64}:
 1.0        0.867481   0.740397   0.621414  …  0.00760248   0.00606525   0.00557377
 0.0257271  0.0250944  0.0235272  0.021334     0.000318206  0.000153182  3.20298e-33

more89

2×21 Matrix{Float64}:
 1.0       0.963161  0.928729  0.896619  …  0.651348   0.648916   0.648106
 0.409894  0.408644  0.404986  0.399006     0.0729363  0.0372793  7.61788e-31

2×21 Matrix{Float64}:
 1.0       0.963161  0.928729  0.896619  …  0.651348   0.648916   0.648106
 0.409894  0.408644  0.404986  0.399006     0.0729363  0.0372793  7.61788e-31

2×21 Matrix{Float64}:
 1.0       0.963161  0.928729  0.896619  …  0.651348   0.648916   0.648106
 0.409894  0.408644  0.404986  0.399006     0.0729363  0.0372793  7.61788e-31

2×21 Matrix{Float64}:
 1.0       0.963161  0.928729  0.896619  …  0.651348   0.648916   0.648106
 0.409894  0.408644  0.404986  0.399006     0.0729363  0.0372793  7.61788e-31

2×21 Matrix{Float64}:
 1.0       0.963161  0.928729  0.896619  …  0.651348   0.648916   0.648106
 0.409894  0.408644  0.404986  0.399006     0.0729363  0.0372793  7.61788e-31

2×21 Matrix{Float64}:
 1.0       0.963161  0.928729  0.896619  …  0.651348   0.648916   0.648106
 0.409894  0.408644  0.404986  0.399006     0.0729363  0.0372793  7.61788e-31

2×21 Matrix{Float64}:
 1.0       0.963161  0.928729  0.896619  …  0.651348   0.648916   0.648106
 0.409894  0.408644  0.404986  0.399006     0.0729363  0.0372793  7.61788e-31

2×21 Matrix{Float64}:
 1.0       0.963161  0.928729  0.896619  …  0.651348   0.648916   0.648106
 0.409894  0.408644  0.404986  0.399006     0.0729363  0.0372793  7.61788e-31

2×21 Matrix{Float64}:
 1.0       0.963161  0.928729  0.896619  …  0.651348   0.648916   0.648106
 0.409894  0.408644  0.404986  0.399006     0.0729363  0.0372793  7.61788e-31

2×21 Matrix{Float64}:
 1.0       0.963161  0.928729  0.896619  …  0.651348   0.648916   0.648106
 0.409894  0.408644  0.404986  0.399006     0.0729363  0.0372793  7.61788e-31

grid

ExtendableGrids.ExtendableGrid{Float64, Int32};
dim: 1 nodes: 21 cells: 20 bfaces: 2

xxxxxxxxxx
 
tsol_readiff=transient_reaction_diffusion(grid1d_a,k=1,tend=100)

1.4 s

time=

27.0 ms

xxxxxxxxxx
 
let
    vis=GridVisualizer(layout=(1,2),resolution=(600,300),Plotter=PyPlot)
    scalarplot!(vis[1,1],tsol_readiff.grid,
           tsol_readiff.solutions[t_readiff][1,:],
           title="u1: t=$(tsol_readiff.times[t_readiff])",
           flimits=(0,1),colormap=:cool,levels=50,clear=true)
    scalarplot!(vis[1,2],tsol_readiff.grid,
           tsol_readiff.solutions[t_readiff][2,:],
           title="u2: t=$(tsol_readiff.times[t_readiff])",
           flimits=(0,1),colormap=:cool,levels=50,show=true)
end
​

326 ms

xxxxxxxxxx
 
TableOfContents()

43.0 ns

xxxxxxxxxx
 
begin 
    ENV["LANG"]="C"
    using Pkg
    Pkg.activate(mktempdir())
    Pkg.add(["PyPlot","PlutoUI","ExtendableGrids","SimplexGridFactory","VoronoiFVM","GridVisualize","Triangulate"])
    using PlutoUI,PyPlot,ExtendableGrids,SimplexGridFactory,VoronoiFVM,GridVisualize,Triangulate
    PyPlot.svg(true)
end;

16.1 s

      Status `/tmp/jl_SKqLWP/Project.toml`
  [cfc395e8] ExtendableGrids v0.7.4
  [5eed8a63] GridVisualize v0.1.3
  [7f904dfe] PlutoUI v0.7.2
  [d330b81b] PyPlot v2.9.0
  [57bfcd06] SimplexGridFactory v0.5.1
  [f7e6ffb2] Triangulate v1.0.1
  [82b139dc] VoronoiFVM v0.10.5

559 ms