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genkuroki/DifferentialEquations.jl example - single pendulum.ipynb

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DifferentialEquations.jl example - single pendulum.ipynb
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DifferentialEquations.jl example - single pendulum.ipynb
{
"cells": [
{
"metadata": {
"toc": true
},
"cell_type": "markdown",
"source": "<h1>Table of Contents<span class=\"tocSkip\"></span></h1>\n<div class=\"toc\"><ul class=\"toc-item\"></ul></div>"
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "VERSION",
"execution_count": 1,
"outputs": [
{
"data": {
"text/plain": "v\"1.7.0-DEV.1129\""
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "using Test\nusing DifferentialEquations\nusing SimpleDiffEq\nusing Plots\nusing Elliptic",
"execution_count": 2,
"outputs": []
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "function my_RK4(f!, u0, tspan, p; dt=0.1)\n t = range(tspan...; step=dt)\n n = length(t)\n u = similar(typeof(u0)[], n)\n u[1] = u0\n du1, du2, du3, du4 = similar(u0), similar(u0), similar(u0), similar(u0)\n dt_half = dt/2\n for k in 2:n\n u_prev, t_prev = u[k-1], t[k-1]\n f!(du1, u_prev , p, t_prev )\n f!(du2, u_prev + du1*dt_half, p, t_prev + dt_half)\n f!(du3, u_prev + du2*dt_half, p, t_prev + dt_half)\n f!(du4, u_prev + du3*dt , p, t_prev + dt )\n u_next = @. u_prev + (du1 + 2du2 + 2du3 + du4)*dt/6\n u[k] = u_next\n end\n (; t, u)\nend",
"execution_count": 3,
"outputs": [
{
"data": {
"text/plain": "my_RK4 (generic function with 1 method)"
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "sn(u, k) = Jacobi.sn(u, k^2)\nsol_exact(p, t) = ((g, l, ymax) = p; 2asin(ymax*sn(√(g/l)*t, ymax)))\n\nfunction f!(du, u, p, t)\n g, l, = p\n du[1] = -(g/l)*sin(u[2])\n du[2] = u[1]\nend\n\np = (g = 9.80665, l = 1.0, ymax = 0.999)\nU0(p) = ((g, l, ymax) = p; [√(g/l)*2ymax, 0.0])\ntspan = (0.0, 30.0)",
"execution_count": 4,
"outputs": [
{
"data": {
"text/plain": "(0.0, 30.0)"
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "dt = 0.1\nsol_fixedstep_RK4 = solve(ODEProblem(f!, U0(p), tspan, p), RK4(); adaptive=false, dt)\nsol_SimpleRK4 = solve(ODEProblem(f!, U0(p), tspan, p), SimpleRK4(); dt)\nsol_my_RK4 = my_RK4(f!, U0(p), tspan, p; dt)\n\n@test sol_fixedstep_RK4.u ≈ sol_SimpleRK4.u ≈ sol_my_RK4.u",
"execution_count": 5,
"outputs": [
{
"data": {
"text/plain": "\u001b[32m\u001b[1mTest Passed\u001b[22m\u001b[39m"
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "E(v, u, p, t) = ((g, l,) = p; v[1]^2/2 - (g/l)*(cos(u[1]) - 1)) # total energy\n\nfunction plot_pendulum(sol; title=\"solution: θ(t)\", ylim=nothing)\n tspan, p = sol.prob.tspan, sol.prob.p\n t = range(tspan..., length=501)\n dθ = (t -> sol(t)[1]).(t)\n θ = (t -> sol(t)[2]).(t)\n θ_exact = sol_exact.(Ref(p), t)\n \n P = plot(t, [θ θ_exact]; label=[\"numerical\" \"exact\"], ls=:auto, ylim=(-4, 6.5))\n plot!(xtick=0:2:30, ytick=-10:10)\n title!(title; titlefontsize=10)\n \n v0, u0 = first.((dθ, θ))\n E_exact = fill(E(v0, u0, p, 0.0), length(t))\n E_numerical = E.(dθ, θ, Ref(p), t)\n Q = plot(t, E_numerical - E_exact; label=\"\")\n title!(\"total energy error\"; titlefontsize=12)\n plot!(xtick=0:2:30)\n isnothing(ylim) || plot!(; ylim)\n \n plot(P, Q; size=(800, 250), layout=(1, 2))\nend",
"execution_count": 6,
"outputs": [
{
"data": {
"text/plain": "plot_pendulum (generic function with 1 method)"
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "dt = 0.1\nsol = solve(ODEProblem(f!, U0(p), tspan, p), RK4(); adaptive=false, dt)\n@show length(sol.t)\nplot_pendulum(sol; title=\"RK4 with fixed dt = $dt\")",
"execution_count": 7,
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": "length(sol.t) = 301\n"
},
{
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},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "dt = 0.05\nsol = solve(ODEProblem(f!, U0(p), tspan, p), RK4(); adaptive=false, dt)\n@show length(sol.t)\nplot_pendulum(sol; title=\"RK4 with fixed dt = $dt\")",
"execution_count": 8,
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": "length(sol.t) = 601\n"
},
{
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2942.33,689.543 2944.83,773.054 2947.34,732.252 2949.84,703.876 2952.35,709.016 2954.85,675.234 2957.36,647.455 2959.86,645.946 2962.37,652.702 \n 2964.87,654.004 2967.38,650.641 2969.88,649.078 2972.39,649.078 2974.89,649.39 2977.4,649.477 2979.9,649.345 2982.41,649.257 2984.91,649.266 2987.42,649.337 \n 2989.92,649.392 2992.43,649.352 2994.93,649.302 2997.44,649.311 2999.94,649.379 3002.45,649.434 3004.95,649.395 3007.46,649.346 3009.96,649.356 3012.47,649.426 \n 3014.98,649.487 3017.48,649.446 3019.99,649.377 3022.49,649.384 3025,649.573 3027.5,649.921 3030.01,649.738 3032.51,648.841 3035.02,648.771 3037.52,653.138 \n 3040.03,662.054 3042.53,658.465 3045.04,644.89 3047.54,652.404 3050.05,693.902 3052.55,711.895 3055.06,739.543 3057.56,856.68 3060.07,887.501 3062.57,763.499 \n 3065.08,709.145 3067.58,747.63 3070.09,722.641 3072.59,705.977 3075.1,721.098 3077.6,726.574 3080.11,709.546 3082.61,700.285 3085.12,700.152 3087.62,701.954 \n 3090.13,702.281 3092.63,701.53 3095.14,701.173 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},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "sol = solve(ODEProblem(f!, U0(p), tspan, p), RK4())\n@show length(sol.t)\nplot_pendulum(sol; title=\"adaptive RK4\")",
"execution_count": 9,
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": "length(sol.t) = 287\n"
},
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},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "dtmax = 0.05\nsol = solve(ODEProblem(f!, U0(p), tspan, p), RK4(); dtmax)\n@show length(sol.t)\nplot_pendulum(sol; title=\"adaptive RK4 with dtmax = $dtmax\")",
"execution_count": 10,
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": "length(sol.t) = 606\n"
},
{
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},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "f1!(du, v, u, p, t) = ((g, l,) = p; du[1] = v[1])\nf2!(dv, v, u, p, t) = ((g, l,) = p; dv[1] = -(g/l)*sin(u[1]))\n\nh(v, u, p, t) = ((g, l,) = p; v[1]^2/2 - (g/l)*(cos(u[1]) + 1)) # Hamiltonian\n\nv0(p) = ((g, l, ymax) = p; [√(g/l)*2ymax])\nu0(p) = ((g, l, ymax) = p; [0.0])",
"execution_count": 11,
"outputs": [
{
"data": {
"text/plain": "u0 (generic function with 1 method)"
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "dtmax = 0.05\nsol = solve(ODEProblem(f!, U0(p), tspan, p); dtmax)\n@show length(sol.t)\nplot_pendulum(sol; title=\"ODEProblem with dtmax = $dtmax\")",
"execution_count": 12,
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": "length(sol.t) = 604\n"
},
{
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517.211,386.677 519.66,393.898 522.11,402.57 524.559,412.945 527.008,425.294 529.457,439.896 531.906,457.006 534.355,476.809 \n 536.804,499.363 539.253,524.518 541.703,551.86 544.152,580.692 546.601,610.091 549.05,639.039 551.499,666.595 553.948,692.029 556.397,714.896 558.846,735.018 \n 561.295,752.432 563.745,767.312 566.194,779.908 568.643,790.497 571.092,799.353 573.541,806.729 575.99,812.853 578.439,817.92 580.888,822.098 583.338,825.528 \n 585.787,828.328 588.236,830.596 590.685,832.408 593.134,833.831 595.583,834.912 598.032,835.69 600.481,836.193 602.931,836.438 605.38,836.434 607.829,836.181 \n 610.278,835.67 612.727,834.882 615.176,833.791 617.625,832.357 620.074,830.531 622.523,828.248 624.973,825.43 627.422,821.977 629.871,817.774 632.32,812.676 \n 634.769,806.516 637.218,799.096 639.667,790.19 642.116,779.541 644.566,766.877 647.015,751.92 649.464,734.423 651.913,714.214 654.362,691.261 656.811,665.751 \n 659.26,638.139 661.709,609.161 664.158,579.765 666.608,550.967 669.057,523.685 671.506,498.607 673.955,476.14 676.404,456.423 678.853,439.396 681.302,424.87 \n 683.751,412.587 686.201,402.27 688.65,393.648 691.099,386.47 693.548,380.514 695.997,375.588 698.446,371.528 700.895,368.197 703.344,365.481 705.793,363.286 \n 708.243,361.534 710.692,360.165 713.141,359.13 715.59,358.394 718.039,357.93 720.488,357.722 722.937,357.762 725.386,358.053 727.836,358.603 730.285,359.434 \n 732.734,360.573 735.183,362.061 737.632,363.95 740.081,366.307 742.53,369.212 744.979,372.767 747.429,377.093 749.878,382.336 752.327,388.668 754.776,396.292 \n 757.225,405.439 759.674,416.367 762.123,429.352 764.572,444.669 767.021,462.558 769.471,483.174 771.92,506.521 774.369,532.377 776.818,560.242 779.267,589.345 \n 781.716,618.716 784.165,647.344 786.614,674.338 789.064,699.05 791.513,721.115 793.962,740.427 796.411,757.071 798.86,771.25 801.309,783.225 803.758,793.276 \n 806.207,801.67 808.656,808.655 811.106,814.448 813.555,819.237 816.004,823.181 818.453,826.414 820.902,829.048 823.351,831.174 825.8,832.866 828.249,834.183 \n 830.699,835.171 833.148,835.866 835.597,836.292 838.046,836.463 840.495,836.386 842.944,836.058 845.393,835.468 847.842,834.594 850.292,833.406 852.741,831.862 \n 855.19,829.909 857.639,827.477 860.088,824.483 862.537,820.822 864.986,816.371 867.435,810.979 869.884,804.469 872.334,796.636 874.783,787.244 877.232,776.03 \n 879.681,762.718 882.13,747.036 884.579,728.752 887.028,707.729 889.477,683.992 891.927,657.802 894.376,629.699 896.825,600.493 899.274,571.169 901.723,542.731 \n 904.172,516.039 906.621,491.702 909.07,470.041 911.519,451.131 913.969,434.865 916.418,421.028 918.867,409.354 921.316,399.564 923.765,391.392 926.214,384.596 \n 928.663,378.962 931.112,374.308 933.562,370.476 936.011,367.337 938.46,364.784 940.909,362.726 943.358,361.093 945.807,359.827 948.256,358.884 950.705,358.23 \n 953.155,357.843 955.604,357.708 958.053,357.822 960.502,358.187 962.951,358.818 965.4,359.736 967.849,360.973 970.298,362.574 972.747,364.593 975.197,367.102 \n 977.646,370.187 980.095,373.956 982.544,378.536 984.993,384.081 987.442,390.772 989.891,398.819 992.34,408.463 994.79,419.969 997.239,433.613 999.688,449.666 \n 1002.14,468.348 1004.59,489.778 1007.04,513.899 1009.48,540.412 1011.93,568.734 1014.38,598.022 1016.83,627.277 1019.28,655.506 1021.73,681.882 1024.18,705.839 \n 1026.63,727.093 1029.08,745.604 1031.53,761.496 1033.98,774.997 1036.42,786.376 1038.87,795.911 1041.32,803.866 1043.77,810.478 1046.22,815.956 1048.67,820.481 \n 1051.12,824.203 1053.57,827.249 1056.02,829.724 1058.47,831.715 1060.92,833.291 1063.36,834.507 1065.81,835.405 1068.26,836.019 1070.71,836.368 1073.16,836.466 \n 1075.61,836.316 1078.06,835.912 1080.51,835.241 1082.96,834.278 1085.41,832.99 1087.86,831.332 1090.31,829.245 1092.75,826.657 1095.2,823.478 1097.65,819.598 \n 1100.1,814.886 1102.55,809.185 1105,802.308 1107.45,794.041 1109.9,784.139 1112.35,772.336 1114.8,758.353 1117.25,741.925 1119.69,722.842 1122.14,701.007 \n 1124.59,676.508 1127.04,649.684 1129.49,621.162 1131.94,591.815 1134.39,562.651 1136.84,534.648 1139.29,508.6 1141.74,485.03 1144.19,464.182 1146.63,446.069 \n 1149.08,430.545 1151.53,417.374 1153.98,406.284 1156.43,396.998 1158.88,389.256 1161.33,382.823 1163.78,377.496 1166.23,373.099 1168.68,369.484 1171.13,366.528 \n 1173.58,364.129 1176.02,362.203 1178.47,360.684 1180.92,359.517 1183.37,358.662 1185.82,358.089 1188.27,357.777 1190.72,357.716 1193.17,357.903 1195.62,358.345 \n 1198.07,359.058 1200.52,360.066 1202.96,361.405 1205.41,363.122 1207.86,365.278 1210.31,367.947 1212.76,371.222 1215.21,375.216 1217.66,380.063 1220.11,385.926 \n 1222.56,392.993 1225.01,401.485 1227.46,411.649 1229.9,423.756 1232.35,438.084 1234.8,454.893 1237.25,474.379 1239.7,496.617 1242.15,521.487 1244.6,548.607 \n 1247.05,577.311 1249.5,606.696 1251.95,635.747 1254.4,663.506 1256.85,689.214 1259.29,712.391 1261.74,732.833 1264.19,750.552 1266.64,765.713 1269.09,778.56 \n 1271.54,789.366 1273.99,798.409 1276.44,805.945 1278.89,812.202 1281.34,817.382 1283.79,821.655 1286.23,825.166 1288.68,828.034 1291.13,830.358 1293.58,832.22 \n 1296.03,833.684 1298.48,834.803 1300.93,835.615 1303.38,836.148 1305.83,836.423 1308.28,836.448 1310.73,836.223 1313.17,835.743 1315.62,834.988 1318.07,833.934 \n 1320.52,832.542 1322.97,830.764 1325.42,828.538 1327.87,825.786 1330.32,822.412 1332.77,818.302 1335.22,813.316 1337.67,807.288 1340.12,800.025 1342.56,791.303 \n 1345.01,780.87 1347.46,768.453 1349.91,753.775 1352.36,736.582 1354.81,716.691 1357.26,694.051 1359.71,668.819 1362.16,641.417 1364.61,612.551 1367.06,583.152 \n 1369.5,554.234 1371.95,526.736 1374.4,501.378 1376.85,478.597 1379.3,458.562 1381.75,441.232 1384.2,426.429 1386.65,413.901 1389.1,403.371 1391.55,394.566 \n 1394,387.233 1396.45,381.146 1398.89,376.11 1401.34,371.958 1403.79,368.549 1406.24,365.767 1408.69,363.515 1411.14,361.716 1413.59,360.305 1416.04,359.234 \n 1418.49,358.465 \n \"/>\n<polyline 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316.384,658.665 318.833,684.784 321.282,708.438 323.731,729.373 326.18,747.572 328.629,763.175 331.078,776.416 333.527,787.568 335.977,796.907 338.426,804.695 \n 340.875,811.166 343.324,816.525 345.773,820.95 348.222,824.587 350.671,827.562 353.12,829.978 355.569,831.917 358.019,833.449 360.468,834.626 362.917,835.49 \n 365.366,836.072 367.815,836.392 370.264,836.461 372.713,836.282 375.162,835.848 377.612,835.145 380.061,834.146 382.51,832.818 384.959,831.114 387.408,828.973 \n 389.857,826.322 392.306,823.067 394.755,819.099 397.205,814.281 399.654,808.453 402.103,801.427 404.552,792.984 407.001,782.877 409.45,770.836 411.899,756.583 \n 414.348,739.857 416.797,720.458 419.247,698.306 421.696,673.515 424.145,646.458 426.594,617.792 429.043,588.413 431.492,559.336 433.941,531.523 436.39,505.741 \n 438.84,482.478 441.289,461.95 443.738,444.145 446.187,428.906 448.636,415.991 451.085,405.123 453.534,396.029 455.983,388.449 458.432,382.154 460.882,376.943 \n 463.331,372.643 465.78,369.111 468.229,366.224 470.678,363.884 473.127,362.008 475.576,360.532 478.025,359.403 480.475,358.582 482.924,358.04 485.373,357.757 \n 487.822,357.725 490.271,357.941 492.72,358.413 495.169,359.158 497.618,360.203 500.068,361.583 502.517,363.348 504.966,365.559 507.415,368.293 509.864,371.645 \n 512.313,375.73 514.762,380.686 517.211,386.677 519.66,393.898 522.11,402.57 524.559,412.945 527.008,425.294 529.457,439.896 531.906,457.006 534.355,476.809 \n 536.804,499.363 539.253,524.518 541.703,551.86 544.152,580.692 546.601,610.091 549.05,639.039 551.499,666.594 553.948,692.029 556.397,714.896 558.846,735.018 \n 561.295,752.432 563.745,767.312 566.194,779.908 568.643,790.497 571.092,799.353 573.541,806.729 575.99,812.853 578.439,817.92 580.888,822.098 583.338,825.528 \n 585.787,828.328 588.236,830.596 590.685,832.408 593.134,833.831 595.583,834.912 598.032,835.69 600.481,836.193 602.931,836.438 605.38,836.434 607.829,836.181 \n 610.278,835.67 612.727,834.882 615.176,833.791 617.625,832.357 620.074,830.531 622.523,828.248 624.973,825.43 627.422,821.977 629.871,817.774 632.32,812.676 \n 634.769,806.516 637.218,799.096 639.667,790.19 642.116,779.541 644.566,766.877 647.015,751.92 649.464,734.423 651.913,714.214 654.362,691.262 656.811,665.751 \n 659.26,638.139 661.709,609.162 664.158,579.765 666.608,550.967 669.057,523.685 671.506,498.607 673.955,476.14 676.404,456.423 678.853,439.397 681.302,424.87 \n 683.751,412.587 686.201,402.27 688.65,393.648 691.099,386.47 693.548,380.514 695.997,375.588 698.446,371.528 700.895,368.197 703.344,365.481 705.793,363.286 \n 708.243,361.534 710.692,360.165 713.141,359.13 715.59,358.394 718.039,357.93 720.488,357.722 722.937,357.762 725.386,358.053 727.836,358.603 730.285,359.434 \n 732.734,360.573 735.183,362.061 737.632,363.95 740.081,366.306 742.53,369.212 744.979,372.767 747.429,377.093 749.878,382.336 752.327,388.668 754.776,396.292 \n 757.225,405.439 759.674,416.367 762.123,429.352 764.572,444.669 767.021,462.557 769.471,483.173 771.92,506.52 774.369,532.376 776.818,560.242 779.267,589.344 \n 781.716,618.716 784.165,647.344 786.614,674.338 789.064,699.049 791.513,721.114 793.962,740.427 796.411,757.071 798.86,771.25 801.309,783.225 803.758,793.276 \n 806.207,801.67 808.656,808.655 811.106,814.448 813.555,819.237 816.004,823.181 818.453,826.414 820.902,829.048 823.351,831.174 825.8,832.866 828.249,834.183 \n 830.699,835.171 833.148,835.866 835.597,836.292 838.046,836.463 840.495,836.386 842.944,836.058 845.393,835.468 847.842,834.594 850.292,833.406 852.741,831.862 \n 855.19,829.909 857.639,827.477 860.088,824.483 862.537,820.822 864.986,816.371 867.435,810.979 869.884,804.47 872.334,796.637 874.783,787.244 877.232,776.03 \n 879.681,762.719 882.13,747.037 884.579,728.753 887.028,707.73 889.477,683.993 891.927,657.803 894.376,629.7 896.825,600.495 899.274,571.17 901.723,542.732 \n 904.172,516.04 906.621,491.703 909.07,470.042 911.519,451.132 913.969,434.866 916.418,421.029 918.867,409.354 921.316,399.565 923.765,391.393 926.214,384.596 \n 928.663,378.963 931.112,374.308 933.562,370.476 936.011,367.337 938.46,364.784 940.909,362.726 943.358,361.093 945.807,359.827 948.256,358.884 950.705,358.23 \n 953.155,357.843 955.604,357.708 958.053,357.822 960.502,358.187 962.951,358.818 965.4,359.736 967.849,360.973 970.298,362.573 972.747,364.593 975.197,367.102 \n 977.646,370.187 980.095,373.956 982.544,378.536 984.993,384.081 987.442,390.771 989.891,398.819 992.34,408.463 994.79,419.968 997.239,433.613 999.688,449.665 \n 1002.14,468.347 1004.59,489.776 1007.04,513.897 1009.48,540.411 1011.93,568.732 1014.38,598.02 1016.83,627.275 1019.28,655.505 1021.73,681.881 1024.18,705.837 \n 1026.63,727.092 1029.08,745.603 1031.53,761.495 1033.98,774.996 1036.42,786.375 1038.87,795.911 1041.32,803.865 1043.77,810.478 1046.22,815.956 1048.67,820.481 \n 1051.12,824.202 1053.57,827.248 1056.02,829.724 1058.47,831.715 1060.92,833.291 1063.36,834.507 1065.81,835.405 1068.26,836.019 1070.71,836.368 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},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "dtmax = 0.05\nsol = solve(DynamicalODEProblem(f2!, f1!, v0(p), u0(p), tspan, p); dtmax)\n@show length(sol.t)\nplot_pendulum(sol; title=\"DynamicalODEProblem with dtmax = $dtmax\")",
"execution_count": 13,
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": "length(sol.t) = 604\n"
},
{
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},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "dtmax = 0.05\nsol = solve(SecondOrderODEProblem(f2!, v0(p), u0(p), tspan, p); dtmax)\n@show length(sol.t)\nplot_pendulum(sol; title=\"SecondOrderODEProblem with dtmax = $dtmax\")",
"execution_count": 14,
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": "length(sol.t) = 604\n"
},
{
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