Add premades

Author: ChrisRackauckas

REQUIRE

@@ -1 +1,6 @@
 julia 0.5
+ParameterizedFunctions
+OrdinaryDiffEq
+StochasticDiffEq
+FiniteElementDiffEq
+JLD

src/DiffEqProblemLibrary.jl

@@ -1,5 +1,38 @@
 module DiffEqProblemLibrary
 
-# package code goes here
+using OrdinaryDiffEq, StochasticDiffEq, ParameterizedFunctions,
+      FiniteElementDiffEq, AlgebraicDiffEq, JLD
+include("ode_premade_problems.jl")
+include("dae_premade_problems.jl")
+include("sde_premade_problems.jl")
+include("fem_premade_problems.jl")
+include("premade_meshes.jl")
+
+#ODE Example Problems
+export prob_ode_linear, prob_ode_bigfloatlinear, prob_ode_2Dlinear,
+       prob_ode_large2Dlinear, prob_ode_bigfloat2Dlinear, prob_ode_rigidbody,
+       prob_ode_2Dlinear_notinplace, prob_ode_vanderpol, prob_ode_vanderpol_stiff,
+       prob_ode_lorenz, prob_ode_rober, prob_ode_threebody
+
+ #SDE Example Problems
+ export prob_sde_wave, prob_sde_linear, prob_sde_cubic, prob_sde_2Dlinear, prob_sde_lorenz,
+        prob_sde_2Dlinear, prob_sde_additive, prob_sde_additivesystem, oval2ModelExample
+
+ #DAE Example Problems
+ export prob_dae_resrob
+
+ #FEM Example Problems
+ export  prob_femheat_moving, prob_femheat_pure, prob_femheat_diffuse,
+         prob_poisson_wave, prob_poisson_noisywave, prob_femheat_birthdeath,
+         prob_poisson_birthdeath, prob_femheat_stochasticbirthdeath,
+         prob_stokes_homogenous, prob_stokes_dirichletzero, prob_poisson_birthdeathsystem,
+         prob_poisson_birthdeathinteractingsystem, prob_femheat_birthdeathinteractingsystem,
+         prob_femheat_birthdeathsystem, prob_femheat_diffusionconstants,
+         heatProblemExample_grayscott,heatProblemExample_gierermeinhardt
+
+ #Example Meshes
+ export  meshExample_bunny, meshExample_flowpastcylindermesh, meshExample_lakemesh,
+         meshExample_Lshapemesh, meshExample_Lshapeunstructure, meshExample_oilpump,
+         meshExample_wavymesh, meshExample_wavyperturbmesh
 
 end # module

src/dae_premade_problems.jl

@@ -0,0 +1,12 @@
+### DAE Problems
+
+f = function (tres, y, yp, r)
+    r[1]  = -0.04*y[1] + 1.0e4*y[2]*y[3]
+    r[2]  = -r[1] - 3.0e7*y[2]*y[2] - yp[2]
+    r[1] -=  yp[1]
+    r[3]  =  y[1] + y[2] + y[3] - 1.0
+end
+u₀ = [1.0, 0, 0]
+du₀ = [-0.04, 0.04, 0.0]
+"DAE residual form for the Robertson model"
+prob_dae_resrob = DAEProblem(f,u₀,du₀)

src/fem_premade_problems.jl

@@ -0,0 +1,183 @@
+### Finite Element Examples
+
+analytic_moving(t,x) = 0.1*(1-exp.(-100*(t-0.5).^2)).*exp.(-25((x[:,1]-t+0.5).^2 + (x[:,2]-t+0.5).^2))
+Du = (t,x) -> -50[analytic_moving(t,x).*(0.5-t+x[:,1])  analytic_moving(t,x).*(0.5-t+x[:,2])]
+f = (t,x) -> (-5).*exp.((-25).*((3/2)+6.*t.^2+x[:,1]+x[:,1].^2+x[:,2]+x[:,2].^2+(-2).*t.*(3+x[:,1]+
+  x[:,2]))).*((-20)+(-100).*t.^2+(-49).*x[:,1]+(-50).*x[:,1].^2+(-49).*x[:,2]+(-50).*
+  x[:,2].^2+2.*t.*(47+50.*x[:,1]+50.*x[:,2])+exp.(25.*(1+(-2).*t).^2).*(22+
+  100.*t.^2+49.*x[:,1]+50.*x[:,1].^2+49.*x[:,2]+50.*x[:,2].^2+(-2).*t.*(49+50.*x[:,1]+50.*x[:,2])))
+"""
+Example problem defined by the solution:
+```math
+u(x,y,t)=\\frac{1}{10}(1-\\exp(-100(t-\\frac{1}{2})^2))\\exp(-25((x-t+0.5)^2 + (y-t+0.5)^2))
+```
+
+This will have a mound which moves across the screen. Good animation test.
+"""
+prob_femheat_moving = HeatProblem(analytic_moving,Du,f)
+
+
+
+analytic_diffuse(t,x) = exp.(-10((x[:,1]-.5).^2 + (x[:,2]-.5).^2 )-t)
+f = (t,x) -> exp.(-t-5*(1-2x[:,1]+2x[:,1].^2 - 2x[:,2] +2x[:,2].^2)).*(-161 + 400*(x[:,1] - x[:,1].^2 + x[:,2] - x[:,2].^2))
+Du = (t,x) -> -20[analytic_diffuse(t,x).*(x[:,1]-.5) analytic_diffuse(t,x).*(x[:,2]-.5)]
+"""
+Example problem defined by the solution:
+
+```math
+u(x,y,t)=\\exp(-10((x-\\frac{1}{2})^2 + (y-\\frac{1}{2})^2 )-t)
+```
+
+This is a Gaussian centered at ``(\\frac{1}{2},\\frac{1}{2})`` which diffuses over time.
+"""
+prob_femheat_diffuse = HeatProblem(analytic_diffuse,Du,f)
+
+
+f = (t,x)  -> zeros(size(x,1))
+u₀ = (x) -> float((abs.(x[:,1]-.5) .< 1e-6) & (abs.(x[:,2]-.5) .< 1e-6)) #Only mass at middle of (0,1)^2
+"""
+Example problem which starts with a Dirac δ cenetered at (0.5,0.5) and solves with ``f=gD=0``.
+This gives the Green's function solution.
+"""
+prob_femheat_pure = HeatProblem(u₀,f)
+
+
+f = (t,x,u) -> ones(size(x,1)) - .5u
+u₀ = (x) -> zeros(size(x,1))
+"""
+Homogenous reaction-diffusion problem which starts with 0 and solves with ``f(u)=1-u/2``
+"""
+prob_femheat_birthdeath = HeatProblem(u₀,f)
+
+
+f = (t,x,u)  -> [ones(size(x,1))-.5u[:,1]   ones(size(x,1))-u[:,2]]
+u₀ = (x) -> ones(size(x,1),2).*[.5 .5] # size (x,2), 2 meaning 2 variables
+"""
+Homogenous reaction-diffusion which starts at 1/2 and solves the system ``f(u)=1-u/2`` and ``f(v)=1-v``
+"""
+prob_femheat_birthdeathsystem = HeatProblem(u₀,f)
+
+f = (t,x,u)  -> [zeros(size(x,1))    zeros(size(x,1))]
+u₀ = (x) -> [float((abs.(x[:,1]-.5) .< 1e-6) & (abs.(x[:,2]-.5) .< 1e-6)) float((abs.(x[:,1]-.5) .< 1e-6) & (abs.(x[:,2]-.5) .< 1e-6))]  # size (x,2), 2 meaning 2 variables
+"""
+Example problem which solves the homogeneous Heat equation with all mass starting at (1/2,1/2) with two different diffusion constants,
+``D₁=0.01`` and ``D₂=0.001``. Good animation test.
+"""
+prob_femheat_diffusionconstants = HeatProblem(u₀,f,D=[.01 .001])
+
+f  = (t,x,u)  -> [ones(size(x,1))-.5u[:,1]     .5u[:,1]-u[:,2]]
+u₀ = (x) -> ones(size(x,1),2).*[.5 .5] # size (x,2), 2 meaning 2 variables
+"""
+Homogenous reaction-diffusion which starts with 1/2 and solves the system ``f(u)=1-u/2`` and ``f(v)=.5u-v``
+"""
+prob_femheat_birthdeathinteractingsystem = HeatProblem(u₀,f)
+
+"""
+`heatProblemExample_grayscott(;ρ=.03,k=.062,D=[1e-3 .5e-3])`
+
+The Gray-Scott equations with quasi-random initial conditions. The reaction equations
+are given by:
+
+```math
+\\begin{align}
+u_t &= uv^2 + ρ(1-v) \\\\
+v_t &= uv^2 - (ρ+k)v
+\\end{align}
+```
+"""
+function heatProblemExample_grayscott(;ρ=.03,k=.062,D=[1e-3 .5e-3])
+  f₁(t,x,u)  = u[:,1].*u[:,2].*u[:,2] + ρ*(1-u[:,2])
+  f₂(t,x,u)  = u[:,1].*u[:,2].*u[:,2] -(ρ+k).*u[:,2]
+  f(t,x,u) = [f₁(t,x,u) f₂(t,x,u)]
+  u₀(x) = [ones(size(x,1))+rand(size(x,1)) .25.*float(((.2.<x[:,1].<.6) &
+          (.2.<x[:,2].<.6)) | ((.85.<x[:,1]) & (.85.<x[:,2])))] # size (x,2), 2 meaning 2 variables
+  return(HeatProblem(u₀,f,D=D))
+end
+
+"""
+`heatProblemExample_gierermeinhardt(;a=1,α=1,D=[0.01 1.0],ubar=1,vbar=0,β=10,startNoise=0.01)`
+
+The Gierer-Meinhardt equations wtih quasi-random initial perturbations.
+
+```math
+\\begin{align}
+u_t &= \\frac{au}{v^2} + \\bar{u} -αu \\\\
+v_t &= au^2 + \\bar{v} - βv
+\\end{align}
+```
+
+The equation starts at the steady state
+
+```math
+\\begin{align}
+u_{ss} &= \\frac{\\bar{u}+β}{α} \\\\
+v_{ss} &= \\frac{α}{β} u_{ss}^2
+\\end{align}
+```
+
+with a bit of noise.
+
+"""
+function heatProblemExample_gierermeinhardt(;a=1,α=1,D=[0.01 1.0],ubar=1,vbar=0,β=10,startNoise=0.01)
+  f₁(t,x,u)  = a*u[:,1].*u[:,1]./u[:,2] + ubar - α*u[:,1]
+  f₂(t,x,u)  = a*u[:,1].*u[:,1] + vbar -β.*u[:,2]
+  f(t,x,u) = [f₁(t,x,u) f₂(t,x,u)]
+  uss = (ubar +β)/α
+  vss = (α/β)*uss.^2
+  u₀(x) = [uss*ones(size(x,1))+startNoise*rand(size(x,1)) vss*ones(size(x,1))] # size (x,2), 2 meaning 2 variables
+  return(HeatProblem(u₀,f,D=D))
+end
+
+f = (t,x,u)  -> ones(size(x,1)) - .5u
+u₀ = (x) -> zeros(size(x,1))
+σ = (t,x,u) -> 1u.^2
+"""
+Homogenous stochastic reaction-diffusion problem which starts with 0
+and solves with ``f(u)=1-u/2`` with noise ``σ(u)=10u^2``
+"""
+prob_femheat_stochasticbirthdeath = HeatProblem(u₀,f,σ=σ)
+
+## Poisson
+
+f = (x) -> sin.(2π.*x[:,1]).*cos.(2π.*x[:,2])
+analytic = (x) -> sin.(2π.*x[:,1]).*cos.(2π.*x[:,2])/(8π*π)
+Du = (x) -> [cos.(2*pi.*x[:,1]).*cos.(2*pi.*x[:,2])./(4*pi) -sin.(2π.*x[:,1]).*sin.(2π.*x[:,2])./(4π)]
+"""
+Problem defined by the solution: ``u(x,y)= \\sin(2πx)\\cos(2πy)/(8π^2)``
+"""
+prob_poisson_wave = PoissonProblem(f,analytic,Du)
+
+σ = (x) -> 5 #Additive noise
+"""
+Problem with deterministic solution: ``u(x,y)= \\sin(2πx)\\cos(2πy)/(8π^2)``
+and additive noise ``σ(x,y)=5``
+"""
+prob_poisson_noisywave = PoissonProblem(f,analytic,Du,σ=σ)
+
+f = (x,u) -> ones(size(x,1)) - .5u
+"""
+Nonlinear Poisson equation with ``f(u)=1-u/2``.
+Corresponds to the steady state of a humogenous reaction-diffusion equation
+with the same ``f``.
+"""
+prob_poisson_birthdeath = PoissonProblem(f,numvars=1)
+
+f  = (x,u) -> [ones(size(x,1))-.5u[:,1]     ones(size(x,1))-u[:,2]]
+u₀ = (x) -> .5*ones(size(x,1),2) # size (x,2), 2 meaning 2 variables
+
+"""
+Nonlinear Poisson equation with ``f(u)=1-u/2`` and ``f(v)=1-v`` and initial
+condition homogenous 1/2. Corresponds to the steady state of a humogenous
+reaction-diffusion equation with the same ``f``.
+"""
+prob_poisson_birthdeathsystem = PoissonProblem(f,u₀=u₀)
+
+f  = (x,u) -> [ones(size(x,1))-.5u[:,1]     .5u[:,1]-u[:,2]]
+u₀ = (x) -> ones(size(x,1),2).*[.5 .5] # size (x,2), 2 meaning 2 variables
+
+"""
+Nonlinear Poisson equation with ``f(u)=1-u/2`` and ``f(v)=.5u-v`` and initial
+condition homogenous 1/2. Corresponds to the steady state of a humogenous
+reaction-diffusion equation with the same ``f``.
+"""
+prob_poisson_birthdeathinteractingsystem = PoissonProblem(f,u₀=u₀)