Merge pull request #145 from AlgebraicJulia/llm/burger

Add Burger's Equation Decapode

examples/diff_adv/burger.jl

@@ -0,0 +1,135 @@
+# AlgebraicJulia Dependencies
+using Decapodes
+using Catlab
+using CombinatorialSpaces
+
+# External Dependencies
+using Logging: global_logger
+using TerminalLoggers: TerminalLogger
+global_logger(TerminalLogger())
+using GeometryBasics: Point2
+Point2D = Point2{Float64}
+using Distributions
+using GLMakie
+using LinearAlgebra
+using MLStyle
+using MultiScaleArrays
+using OrdinaryDiffEq
+
+# Represent component Decapodes.
+Diffusion = @decapode begin
+  C::Form0
+  ϕ::Form1
+  ν::Constant
+
+  # Fick's first law
+  ϕ == ν * d(C)
+end
+
+Advection = @decapode begin
+  C::Form0
+  (V, ϕ)::Form1
+
+  ϕ == ∧₀₁(C,V)
+end
+
+Lie = @decapode begin
+  C::Form0
+  V::Form1
+  dX::Form1
+
+  V == ∘(⋆,⋆)(C ∧ dX)
+end
+
+Superposition = @decapode begin
+  (C, Ċ)::Form0
+  (ϕ, ϕ₁, ϕ₂)::Form1
+
+  ϕ == ϕ₁ + ϕ₂
+  Ċ == ∘(⋆,d,⋆)(ϕ)
+  ∂ₜ(C) == Ċ
+end
+
+# Compose physics.
+compose_burger = @relation () begin
+  diffusion(C, ϕ₁)
+  advection(C, ϕ₂, V)
+  lie(C, V)
+  superposition(ϕ₁, ϕ₂, ϕ, C)
+end
+
+to_graphviz(compose_burger, box_labels=:name, junction_labels=:variable, prog="circo")
+
+Burger_cospan = oapply(compose_burger,
+               [Open(Diffusion,     [:C, :ϕ]),
+                Open(Advection,     [:C, :ϕ, :V]),
+                Open(Lie,           [:C, :V]),
+                Open(Superposition, [:ϕ₁, :ϕ₂, :ϕ, :C])])
+Burger = apex(Burger_cospan)
+
+# Specify semantics of the 1D DEC.
+# i.e. Declare these dynamics are happening on a line.
+Burger = expand_operators(Burger)
+infer_types!(Burger, op1_inf_rules_1D, op2_inf_rules_1D)
+resolve_overloads!(Burger, op1_res_rules_1D, op2_res_rules_1D)
+
+to_graphviz(Burger)
+
+# Create mesh.
+# This is a line. This could be a helper function.
+s = EmbeddedDeltaSet1D{Bool, Point2D}()
+add_vertices!(s, 1000, point=Point2D.(1:1000,0))
+add_edges!(s, 1:(nv(s)-1), 2:nv(s))
+sd = EmbeddedDeltaDualComplex1D{Bool, Float64, Point2D}(s)
+subdivide_duals!(sd, Circumcenter())
+
+# Set initial conditions and constants.
+c_dist = MvNormal([500, 5], [10.5, 10.5])
+c = [pdf(c_dist, [p[1], p[2]]) for p in point(sd)]
+dX = ones(ne(sd))
+
+u₀ = construct(PhysicsState, [VectorForm(c), VectorForm(dX)], Float64[], [:C, :lie_dX])
+
+cs_ps = (diffusion_ν = 0.0005,)
+
+# Describe mappings from symbols to discrete differential operators.
+function generate(sd, my_symbol; hodge=DiagonalHodge())
+  op = @match my_symbol begin
+    # Specify which wedge product to use.
+    # This should probably be the default.
+    :∧₀₁ => (x,y) -> begin
+      ∧(Tuple{0,1},sd,x,y)
+    end
+    x => error("Unmatched operator $my_symbol")
+  end
+  return (args...) -> op(args...)
+end
+
+# Generate simulation.
+sim = eval(gensim(Burger, dimension=1))
+fₘ = sim(sd, generate, DiagonalHodge())
+
+# Run simulation.
+tₑ = 1e5
+prob = ODEProblem(fₘ, u₀, (0.0, tₑ), cs_ps)
+sol = solve(prob, Tsit5(), progress=true, progress_steps=1)
+
+# Visualize initial and final conditions.
+lines(map(x -> x[1], point(sd)), findnode(sol(0.0), :C))
+lines!(map(x -> x[1], point(sd)), findnode(sol(tₑ), :C))
+
+# Animate the dynamics.
+times = range(0.0, tₑ, length=150)
+colors = [findnode(sol(t), :C) for t in times]
+
+frames = 100
+fig = Figure(resolution = (800, 800))
+ax1 = Axis(fig[1,1])
+xlims!(ax1, extrema(map(x -> x[1], point(sd))))
+ylims!(ax1, extrema(findnode(sol(0.0), :C)))
+Label(fig[1,1,Top()], "Speed C")
+Label(fig[2,1,Top()], "Line plot of speed of fluid along the linear domain, every $(tₑ/frames) time units")
+
+record(fig, "burger_low_diff.gif", range(0.0, tₑ; length=frames); framerate = 15) do t
+  lines!(fig[1,1], map(x -> x[1], point(sd)), findnode(sol(t), :C))
+end