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ketch/LV examples.ipynb

Created October 26, 2021 13:26
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LV examples.ipynb
{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"id": "f7063e3b",
"metadata": {
"scrolled": false
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"\u001b[32m\u001b[1m Updating\u001b[22m\u001b[39m registry at `~/.julia/registries/General`\n",
"\u001b[32m\u001b[1m Updating\u001b[22m\u001b[39m git-repo `https://github.com/JuliaRegistries/General.git`\n",
"\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/Research/Projects/2021_bseries_paper/notebooks/Project.toml`\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/Research/Projects/2021_bseries_paper/notebooks/Manifest.toml`\n",
"\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/Research/Projects/2021_bseries_paper/notebooks/Project.toml`\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/Research/Projects/2021_bseries_paper/notebooks/Manifest.toml`\n",
"\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/Research/Projects/2021_bseries_paper/notebooks/Project.toml`\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/Research/Projects/2021_bseries_paper/notebooks/Manifest.toml`\n",
"\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/Research/Projects/2021_bseries_paper/notebooks/Project.toml`\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/Research/Projects/2021_bseries_paper/notebooks/Manifest.toml`\n",
"\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/Research/Projects/2021_bseries_paper/notebooks/Project.toml`\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/Research/Projects/2021_bseries_paper/notebooks/Manifest.toml`\n"
]
}
],
"source": [
"using Pkg; Pkg.add(\"BSeries\")\n",
"import Pkg; Pkg.add(\"LaTeXStrings\")\n",
"import Pkg; Pkg.add(\"Plots\")\n",
"import Pkg; Pkg.add(\"StaticArrays\")\n",
"import Pkg; Pkg.add(\"Symbolics\")\n",
"\n",
"using Pkg; Pkg.update(\"BSeries\")\n",
"import Pkg; Pkg.add(\"OrdinaryDiffEq\")"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "cb4738e2",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"retcode: Success\n",
"Interpolation: specialized 4th order \"free\" interpolation\n",
"t: 26-element Vector{Float64}:\n",
" 0.0\n",
" 0.10588317465376221\n",
" 0.31559982093689176\n",
" 0.6057867045769587\n",
" 0.917388053456856\n",
" 1.27233193166609\n",
" 1.709595465004439\n",
" 2.2710715797715295\n",
" 2.817767101556503\n",
" 3.4887094096806184\n",
" 4.1252644691353675\n",
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" 5.552102465714351\n",
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" 6.967790496422689\n",
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" 8.50025914761285\n",
" 9.252651987137254\n",
" 9.919062712718954\n",
" 10.536704882104361\n",
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" 12.886989585688136\n",
" 13.638667977355583\n",
" 14.315310694703227\n",
" 15.0\n",
"u: 26-element Vector{Vector{Float64}}:\n",
" [1.5, 2.25]\n",
" [1.4517473240580008, 2.3666236836466745]\n",
" [1.3151259674535085, 2.566803961318523]\n",
" [1.0864766945506006, 2.721515660361675]\n",
" [0.8667308450594411, 2.6977572232011022]\n",
" [0.699044690716124, 2.490766480998034]\n",
" [0.6092535820647252, 2.1322734053085317]\n",
" [0.6396027603753301, 1.7150813288436226]\n",
" [0.8069671075854029, 1.4651581617562945]\n",
" [1.1858043706666668, 1.4488810795244689]\n",
" [1.5226353743487921, 1.8403528457329017]\n",
" [1.2018684178316834, 2.6669366441339846]\n",
" [0.7977614272323853, 2.6438332607832074]\n",
" [0.6142317920937794, 2.1804430524441236]\n",
" [0.6803498176055564, 1.6101402515184906]\n",
" [0.9785360728507907, 1.4101209381944313]\n",
" [1.477189575041681, 1.7169006614497406]\n",
" [1.3741713797476944, 2.496703678606309]\n",
" [0.8740408771264914, 2.7052420407654836]\n",
" [0.6328372309023396, 2.290757857341663]\n",
" [0.6452461615584377, 1.6929587802746175]\n",
" [0.9056648799767171, 1.4201475575931688]\n",
" [1.4227856950951772, 1.628430786091787]\n",
" [1.446674260459961, 2.381029455999774]\n",
" [0.9432507640700953, 2.732382248224646]\n",
" [0.6406110221276673, 2.323323328811831]"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using OrdinaryDiffEq\n",
"\n",
"function f(du, u, params, t)\n",
" p, q = u\n",
" dp = (2 - q) * p\n",
" dq = (p - 1) * q\n",
" du[1] = dp; du[2] = dq\n",
" return nothing\n",
"end\n",
"\n",
"u0 = [1.5, 2.25]\n",
"tspan = (0.0, 15.0)\n",
"ode = ODEProblem(f, u0, tspan)\n",
"\n",
"dt = 0.1\n",
"sol_euler = solve(ode, Euler(), dt=dt)\n",
"sol_ref = solve(ode, Tsit5())"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "2fe6abb8",
"metadata": {},
"outputs": [
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]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using LaTeXStrings, Plots\n",
"\n",
"fig = plot(xguide=L\"$q$\", yguide=L\"$p$\")\n",
"default(linewidth=2)\n",
"plot!(fig, sol_ref, vars=(2, 1), label=\"Reference solution\")\n",
"scatter!(fig, last.(sol_euler.u), first.(sol_euler.u),\n",
" label=\"Explicit Euler, dt = $dt\")\n",
"plot!(fig, xlims=(0.0, 9.0), ylims=(0.0, 5.0))"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "39473c59",
"metadata": {},
"outputs": [],
"source": [
"using BSeries, StaticArrays, Symbolics\n",
"\n",
"function solve_modified_equation(ode, truncation_orders, dt)\n",
" # Explicit Euler method\n",
" A = @SArray [0//1;]\n",
" b = @SArray [1//1]\n",
" c = @SArray [0//1]\n",
"\n",
" # Setup of symbolic variables\n",
" @variables dt_sym\n",
" u_sym = @variables p q\n",
" f_sym = similar(u_sym); f(f_sym, u_sym, nothing, nothing)\n",
"\n",
" sol_euler = solve(ode, Euler(), dt=dt)\n",
"\n",
" fig = plot(xguide=L\"$q$\", yguide=L\"$p$\")\n",
" default(linewidth=2)\n",
" scatter!(fig, last.(sol_euler.u), first.(sol_euler.u),\n",
" label=\"Explicit Euler, dt = $dt\")\n",
"\n",
" for truncation_order in truncation_orders\n",
" series = modified_equation(f_sym, u_sym, dt_sym, A, b, c, truncation_order)\n",
" series = Symbolics.substitute.(series, dt_sym => dt)\n",
" modified_f, _ = build_function(series, u_sym, expression=Val(false))\n",
" modified_ode = ODEProblem((u, params, t) -> modified_f(u), ode.u0, tspan)\n",
" modified_sol = solve(modified_ode, Tsit5())\n",
" plot!(fig, modified_sol, vars=(2, 1),\n",
" label=\"Modified ODE, order $(truncation_order-1)\")\n",
" end\n",
" fig\n",
"end\n",
"\n",
"fig = solve_modified_equation(ode, 2, dt)\n",
"savefig(\"../figures/LV1.pdf\")"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "915d9688",
"metadata": {},
"outputs": [],
"source": [
"fig = solve_modified_equation(ode, 2:3, 0.11)\n",
"savefig(\"../figures/LV2.pdf\")"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "f254b85d",
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"┌ Warning: Interrupted. Larger maxiters is needed.\n",
"└ @ SciMLBase /Users/ketch/.julia/packages/SciMLBase/h4Gxc/src/integrator_interface.jl:331\n",
"┌ Warning: Interrupted. Larger maxiters is needed.\n",
"└ @ SciMLBase /Users/ketch/.julia/packages/SciMLBase/h4Gxc/src/integrator_interface.jl:331\n"
]
}
],
"source": [
"fig = solve_modified_equation(ode, 2:4, 0.12)\n",
"savefig(\"../figures/LV3.pdf\")"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "598bd443",
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"┌ Warning: Instability detected. Aborting\n",
"└ @ SciMLBase /Users/ketch/.julia/packages/SciMLBase/h4Gxc/src/integrator_interface.jl:351\n"
]
}
],
"source": [
"u0 = [1., 2.01]\n",
"tspan = (0.0, 66.4)\n",
"ode = ODEProblem(f, u0, tspan)\n",
"\n",
"dt = 0.1\n",
"sol_euler = solve(ode, Euler(), dt=dt)\n",
"sol_ref = solve(ode, Tsit5())\n",
"\n",
"# Explicit Euler method\n",
"A = @SArray [0//1;]\n",
"b = @SArray [1//1]\n",
"c = @SArray [0//1]\n",
"\n",
"# Setup of symbolic variables\n",
"@variables dt_sym\n",
"u_sym = @variables p q\n",
"f_sym = similar(u_sym); f(f_sym, u_sym, nothing, nothing)\n",
"\n",
"sol_euler = solve(ode, Euler(), dt=dt)\n",
"\n",
"fig = plot(xguide=L\"$q$\", yguide=L\"$p$\")\n",
"default(linewidth=2)\n",
"scatter!(fig, last.(sol_euler.u), first.(sol_euler.u),\n",
" label=\"Explicit Euler, dt = $dt\")\n",
"\n",
"for truncation_order in 2:7\n",
" series = modified_equation(f_sym, u_sym, dt_sym, A, b, c, truncation_order)\n",
" series = Symbolics.substitute.(series, dt_sym => dt)\n",
" modified_f, _ = build_function(series, u_sym, expression=Val(false))\n",
" modified_ode = ODEProblem((u, params, t) -> modified_f(u), ode.u0, tspan)\n",
" modified_sol = solve(modified_ode, Tsit5())\n",
" plot!(fig, modified_sol, vars=(2, 1),\n",
" label=\"Modified ODE, order $(truncation_order-1)\")\n",
"end\n",
"fig\n",
"savefig(\"../figures/LV4.pdf\")"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "2bae2530",
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Julia 1.6.3",
"language": "julia",
"name": "julia-1.6"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.6.3"
}
},
"nbformat": 4,
"nbformat_minor": 5
}
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