advection problems

advection.jl

# ]add AbstractPlotting
# ]add Makie

using Makie
function step!(u_new, u, o)
    # u_new[i] = u[i] + dt*v*(u[i] - u[i-1])/dx
    # 
    for i in reverse(eachindex(u))
        s = o.v*o.dt/o.dx
        u_new[i] = u[i] + s * (u[i] - get(u, i-1, zero(eltype(u))))
    end
    u_new
end
function step(u, o)
    u_new = similar(u)
    step!(u_new, u, o)
end

function advectionplot(u0, o)
    t = Node(0.0)
    u = Node(copy(u0))
    display(Makie.plot(u))
    
    map(t) do _
        step!(u.val,u.val,o)
        push!(u, u.val)
    end
    for _ in 1:length(u0)
        sleep(o.dt/length(u0))
        push!(t, 0.)
    end
end

function delta(i, n=1000)
    ret = zeros(n)
    ret[i] = 1
    ret
end

function gauss(i, n=1000)
    ret = map(1:n) do idx
        x = (idx - i) / 50
        exp(-x^2/2)
    end
end



# optimal parameters
o = (dt=1, v=-1, dx=1)
u0 = delta(50)
u0 = gauss(500)
advectionplot(u0, o)

# upwind/downwind wrong

o = (dt=5, v=1, dx=5)
u0 = gauss(500)
advectionplot(u0, o)

# numerical diffusion:
# dx too large
o = (dt=1, v=-1, dx=1.1)
u0 = gauss(500)
u0 = delta(500)
advectionplot(u0, o)

# instability:
# dx too small
o = (dt=1, v=-1, dx=0.9)
u0 = gauss(500)
advectionplot(u0, o)