Burgers' equation resolved by FEM

example/Burgers_FEM/Project.toml

@@ -0,0 +1,17 @@
+name = "Burgers_FEM"
+uuid = "e5170f7b-8a26-429b-971b-688deff48327"
+authors = ["Yueh-Hua Tu"]
+version = "0.1.0"
+
+[deps]
+FEniCS = "186dfeec-b415-5c13-8e76-5fbf19f56f9b"
+OrdinaryDiffEq = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed"
+
+[compat]
+julia = "1.6"
+
+[extras]
+Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
+
+[targets]
+test = ["Test"]

example/Burgers_FEM/README.md

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+# Burgers' equation resolved by finite element method
+
+This example exhibits PDE problem for Burgers' equation resolved by finite element method. Change directory to `example/Burgers_FEM` and use following commend to solve the problem:
+
+```julia
+$ julia --proj
+
+julia> using Burgers_FEM; Burgers_FEM.run_fem()
+```

example/Burgers_FEM/src/Burgers_FEM.jl

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+module Burgers_FEM
+
+using FEniCS
+
+function run_fem(ν = 1 / 1000)
+    # parameters
+    s = 1024  # x
+    steps = 200  # t
+
+    DT = Constant(1 / steps)
+    dt = 1 / steps
+
+    mesh = UnitIntervalMesh(s)
+    V = FunctionSpace(mesh, "CG", 1)
+
+    bc = DirichletBC(V, 0.0, "on_boundary")
+
+    u_init = Expression("x[0]", degree = 1)
+    u = TrialFunction(V)
+    u_old = FeFunction(V)
+    v = TestFunction(V)
+
+    u = interpolate(u_init, V)
+    assign(u_old, u)
+
+    f = Expression("0.0", degree = 0)
+
+    F = (dot(u - u_old, v) / DT
+         + ν * inner(grad(u), grad(v))
+         + inner(u * directional_derivative(u, 0), v)
+         -
+         dot(f, v)) * dx
+
+    us = Vector{Float64}[]
+    t = 0.0
+    for n in 1:steps
+        t = t + dt
+        nlvsolve(F, u, bc)
+        push!(us, get_array(u))
+        assign(u_old, u)
+    end
+
+    return us
+end
+
+end

example/Burgers_FEM/test/runtests.jl

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+using Burgers_FEM
+using Test
+
+@testset "Burgers_FEM.jl" begin
+    # Write your tests here.
+end

test/runtests.jl

@@ -27,3 +27,4 @@ end
     　l、ﾞ ~ヽ
     　じしf_, )ノ
 =#
+# Want some fish?