/
BurgersEquation.jl
1436 lines (1179 loc) · 48.6 KB
/
BurgersEquation.jl
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### A Pluto.jl notebook ###
# v0.19.22
using Markdown
using InteractiveUtils
# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
quote
local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
local el = $(esc(element))
global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
el
end
end
# ╔═╡ fa059506-0710-11ec-1bb4-5f0937e3b169
begin
using GradientRobustMultiPhysics
using ExtendableGrids
using SimplexGridFactory
using ForwardDiff
using Triangulate
using GridVisualize
using PlutoVista
default_plotter!(PlutoVista)
using PlutoUI
using Pkg
#Pkg.add(Pkg.PackageSpec(;name="GradientRobustMultiPhysics", rev="master"))
Pkg.status()
end
# ╔═╡ 582e3416-5d15-4d01-936a-bd2a1e6b7310
md"""
## Tutorial notebook: Burger's equation
This example solves the one-dimensional (viscous) Burger's equation
```math
\begin{aligned}
u_t - \mu \Delta u + \mathrm{div} f(u) & = 0
\end{aligned}
```
on the interval ``(-2,2)`` with periodic boundary conditions, ``f(u) = \frac{1}{2} u^2`` and some initial profile ``u(0) = u_0``.
### Problem description
"""
# ╔═╡ ef7b06ec-7614-4048-a50b-90301dd24b32
md"""
### Grid generation
"""
# ╔═╡ 6243a27b-706f-4d4b-acbd-5431868e2b6f
begin
## GRID
h = 0.02
grid = simplexgrid(-2:h:2)
@info grid
end
# ╔═╡ e882b9a6-5804-4381-a09c-bf9df81d798a
begin
## PROBLEM DATA
f = (result, u) -> (result[1] = u[1]^2/2;) # kernel for nonlinearity
u0 = DataFunction((result,x) -> (result[1] = abs(x[1]) < 0.5 ? 1 : 0), [1, 1]; dependencies = "X", bonus_quadorder = 10) # initial state
ν = 1e-2
## PROBLEM DESCRIPTION
Problem = PDEDescription("Burger's Equation")
add_unknown!(Problem; unknown_name = "u", equation_name = "Burger's Equation")
add_operator!(Problem, 1, NonlinearForm(Gradient, [Identity], [1], f, [1,1]; name = "(f(#A),∇#T)", newton = true, bonus_quadorder = 2))
add_operator!(Problem, [1,1], LaplaceOperator(ν))
add_constraint!(Problem, CombineDofs(1, 1, [1], [num_nodes(grid)]))
@info Problem
end
# ╔═╡ 575be54b-e46a-4576-9190-e826d37def7c
md"""
### Finite element spaces
"""
# ╔═╡ 0de84d2f-9371-4352-b4a0-9b446cb297f8
begin
## FINITE ELEMENT SPACES
order = 2
FES = FESpace{H1Pk{1,1,order}}(grid)
sol = FEVector(FES)
@info sol
end
# ╔═╡ d1994d6f-e77f-4a7e-8daa-f60ef8df4dff
md"""
### Solving
"""
# ╔═╡ 0301f52c-7b36-46c5-aea1-958dece2b304
begin
## INITIAL CONDITION
interpolate!(sol[1], u0)
## PREPARE TIME EVOLUTION
τ = 1//10
nsteps = 200
tcs = TimeControlSolver(Problem, sol, BackwardEuler;
timedependent_equations = [1],
maxiterations = 5,
target_residual = 1e-10,
T_time = typeof(τ))
## EVOLVE
nviews_c = nodevalues_view(sol[1])
sol4t = [nviews_c[1][:]] # save solution at all timesteps
for j = 1 : nsteps
advance!(tcs, τ) # single time step advance
# print current time, iterations, nonlinear residuals for each equation
@info tcs.ctime, Int(tcs.statistics[1,4]), tcs.statistics[:,3]
push!(sol4t, nviews_c[1][:])
end
end
# ╔═╡ 0eb8b37f-c08f-4fd2-aa7e-d745336f0ee0
md"""
## Results
"""
# ╔═╡ 90c83f13-d298-45c7-90d9-1739b3e810fb
md"""
Time: $(@bind t Slider(0:length(sol4t)-1,show_value=true,default=0))
"""
# ╔═╡ 136ddff3-82bc-458e-952e-8afc13982cf4
scalarplot(grid, sol4t[t+1]; limits = (-0.1,1.1), levels = 1, title = "c ( T=$(Float64(t*τ)) )")
# ╔═╡ 00000000-0000-0000-0000-000000000001
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