Adding parameters

We can also build up more complicated systems with multiple dependent variables and parameters as follows

using ModelingToolkit, MethodOfLines, OrdinaryDiffEq, DomainSets

@parameters t x
@parameters Dn, Dp
@variables u(..) v(..)
Dt = Differential(t)
Dx = Differential(x)
Dxx = Differential(x)^2

eqs  = [Dt(u(t,x)) ~ Dn * Dxx(u(t,x)) + u(t,x)*v(t,x),
        Dt(v(t,x)) ~ Dp * Dxx(v(t,x)) - u(t,x)*v(t,x)]
bcs = [u(0,x) ~ sin(pi*x/2),
       v(0,x) ~ sin(pi*x/2),
       u(t,0) ~ 0.0, Dx(u(t,1)) ~ 0.0,
       v(t,0) ~ 0.0, Dx(v(t,1)) ~ 0.0]

domains = [t ∈ Interval(0.0,1.0),
           x ∈ Interval(0.0,1.0)]

@named pdesys = PDESystem(eqs,bcs,domains,[t,x],[u(t,x),v(t,x)],[Dn=>0.5, Dp=>2])

discretization = MOLFiniteDifference([x=>0.1],t)

prob = discretize(pdesys,discretization) # This gives an ODEProblem since it's time-dependent

sol = solve(prob,Tsit5())

discrete_x = sol[x]
discrete_t = sol[t]

solu = sol[u(t, x)]
solv = sol[v(t, x)]

using Plots

anim = @animate for i in 1:length(t)
    p1 = plot(discrete_x, solu[i, :], label="u, t=$(discrete_t[i])[1:9] "; legend=false, xlabel="x",ylabel="u",ylim=[0,1])
    p2 = plot(discrete_x, solv[i, :], label="v, t=$(discrete_t[i])"; legend=false, xlabel="x", ylabel="v",ylim=[0, 1])
    plot(p1, p2)
end
gif(anim, "plot.gif",fps=30)

Remake with different parameter values

The system does not need to be re-discretized every time we want to plot with different parameters, the system can be remade with new parameters with remake. See the ModelingToolkit.jldocs for more ways to manipulate a prob post discretization.

using ModelingToolkit, MethodOfLines, OrdinaryDiffEq, DomainSets

@parameters t x
@parameters Dn, Dp
@variables u(..) v(..)
Dt = Differential(t)
Dx = Differential(x)
Dxx = Differential(x)^2

eqs  = [Dt(u(t, x)) ~ Dn * Dxx(u(t, x)) + u(t, x)*v(t,x),
        Dt(v(t, x)) ~ Dp * Dxx(v(t, x)) - u(t, x)*v(t,x)]
bcs = [u(0, x) ~ sin(pi*x/2),
       v(0, x) ~ sin(pi*x/2),
       u(t, 0) ~ 0.0, Dx(u(t, 1)) ~ 0.0,
       v(t, 0) ~ 0.0, Dx(v(t, 1)) ~ 0.0]

domains = [t ∈ Interval(0.0, 1.0),
           x ∈ Interval(0.0, 1.0)]

@named pdesys = PDESystem(eqs, bcs, domains,[t, x], [u(t, x), v(t, x)], [Dn=>0.5, Dp=>2.0])

discretization = MOLFiniteDifference([x=>0.1], t)

prob = discretize(pdesys,discretization) # This gives an ODEProblem since it's time-dependent

sols = []
for (Dnval, Dpval) in zip(rand(10), rand(10))
    newprob = remake(prob, p = [Dnval, Dpval])
    push!(sols, solve(newprob, Tsit5()));
end


using Plots
for (j, sol) in enumerate(sols)
    discrete_x = sol[x]
    discrete_t = sol[t]
    solu = sol[u(t, x)]
    solv = sol[v(t, x)]
    anim = @animate for i in 1:length(discrete_t)
        p1 = plot(discrete_x, solu[i, :], label="u, t=$(discrete_t[i])"; legend=false, xlabel="x",ylabel="u",ylim=[0,1])
        p2 = plot(discrete_x, solv[i, :], label="v, t=$(discrete_t[i])"; legend=false, xlabel="x", ylabel="v",ylim=[0, 1])
        plot(p1, p2)
    end
    gif(anim, "plot_$j.gif",fps=10)
end

┌ Warning: : no method matching get_unit for arguments (Pair{Symbolics.Num, Float64},).
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/66mEN/src/systems/validation.jl:154
┌ Warning: : no method matching get_unit for arguments (Pair{Symbolics.Num, Float64},).
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/66mEN/src/systems/validation.jl:154
┌ Warning: : no method matching get_unit for arguments (Pair{Symbolics.Num, Float64},).
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/66mEN/src/systems/validation.jl:154
┌ Warning: : no method matching get_unit for arguments (Pair{Symbolics.Num, Float64},).
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/66mEN/src/systems/validation.jl:154
┌ Warning: : no method matching get_unit for arguments (Pair{Symbolics.Num, Float64},).
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/66mEN/src/systems/validation.jl:154
┌ Warning: : no method matching get_unit for arguments (Pair{Symbolics.Num, Float64},).
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/66mEN/src/systems/validation.jl:154
[ Info: Saved animation to /home/runner/work/MethodOfLines.jl/MethodOfLines.jl/docs/build/tutorials/plot_1.gif
[ Info: Saved animation to /home/runner/work/MethodOfLines.jl/MethodOfLines.jl/docs/build/tutorials/plot_2.gif
[ Info: Saved animation to /home/runner/work/MethodOfLines.jl/MethodOfLines.jl/docs/build/tutorials/plot_3.gif
[ Info: Saved animation to /home/runner/work/MethodOfLines.jl/MethodOfLines.jl/docs/build/tutorials/plot_4.gif
[ Info: Saved animation to /home/runner/work/MethodOfLines.jl/MethodOfLines.jl/docs/build/tutorials/plot_5.gif
[ Info: Saved animation to /home/runner/work/MethodOfLines.jl/MethodOfLines.jl/docs/build/tutorials/plot_6.gif
[ Info: Saved animation to /home/runner/work/MethodOfLines.jl/MethodOfLines.jl/docs/build/tutorials/plot_7.gif
[ Info: Saved animation to /home/runner/work/MethodOfLines.jl/MethodOfLines.jl/docs/build/tutorials/plot_8.gif
[ Info: Saved animation to /home/runner/work/MethodOfLines.jl/MethodOfLines.jl/docs/build/tutorials/plot_9.gif
[ Info: Saved animation to /home/runner/work/MethodOfLines.jl/MethodOfLines.jl/docs/build/tutorials/plot_10.gif