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Created September 13, 2021 11:54
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{
"cells": [
{
"cell_type": "markdown",
"id": "d087382e",
"metadata": {},
"source": [
"In this notebook, we use the package [`bseries.py`](https://github.com/ketch/bseries) to derive modifying integrators for certain Runge-Kutta methods applied to first-order ODEs, and study how well the solution of the truncated equations approximates the exact solution."
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "75c2aa03",
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"from BSeries import trees, bs\n",
"import matplotlib.pyplot as plt\n",
"from nodepy import rk, ivp\n",
"from IPython.display import display, Math\n",
"import sympy\n",
"from sympy import symbols, simplify, lambdify, dsolve, Eq, Function\n",
"from sympy import Derivative as D\n",
"from sympy.abc import t\n",
"cf = trees.canonical_forest\n",
"one = sympy.Rational(1)\n",
"from sympy import sin\n",
"from scipy.integrate import solve_ivp\n",
"h = sympy.Symbol('h')"
]
},
{
"cell_type": "markdown",
"id": "baae1fb0",
"metadata": {},
"source": [
"# Rigid body Euler equations"
]
},
{
"cell_type": "markdown",
"id": "51545e26",
"metadata": {},
"source": [
"Here we reproduce the first example from CHV2007. First we set up the system of 3 ODEs corresponding to the rigid body problem (CHV2007 eqn. (9)):"
]
},
{
"cell_type": "code",
"execution_count": 70,
"id": "cbe9adae",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[y1, y2, y3]"
]
},
"execution_count": 70,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from sympy.abc import alpha, beta, gamma\n",
"from sympy.abc import x, y, z\n",
"y = [symbols('y%d' % i) for i in range(1,4)]\n",
"\n",
"u = [y[0],y[1],y[2]]\n",
"u"
]
},
{
"cell_type": "code",
"execution_count": 74,
"id": "36a6f75c",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\alpha y_{2} y_{3}$"
],
"text/plain": [
"alpha*y2*y3"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\beta y_{1} y_{3}$"
],
"text/plain": [
"beta*y1*y3"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\gamma y_{1} y_{2}$"
],
"text/plain": [
"gamma*y1*y2"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"f = np.array([alpha*y[1]*y[2],beta*y[2]*y[0],gamma*y[0]*y[1]])\n",
"for rhs in f:\n",
" display(rhs)"
]
},
{
"cell_type": "markdown",
"id": "396366e4",
"metadata": {},
"source": [
"We will derive a modifying integrator ODE system for the implicit midpoint method."
]
},
{
"cell_type": "code",
"execution_count": 75,
"id": "4f8ebf95",
"metadata": {},
"outputs": [],
"source": [
"# Implicit Midpoint Method\n",
"A = np.array([[one/2]])\n",
"b = np.array([one])"
]
},
{
"cell_type": "code",
"execution_count": 76,
"id": "ac221806",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}\\frac{\\alpha y_{2} y_{3} \\left(\\alpha^{2} \\beta^{2} h^{4} y_{3}^{4} + 4 \\alpha^{2} \\beta \\gamma h^{4} y_{2}^{2} y_{3}^{2} + \\alpha^{2} \\gamma^{2} h^{4} y_{2}^{4} + 4 \\alpha \\beta^{2} \\gamma h^{4} y_{1}^{2} y_{3}^{2} + 4 \\alpha \\beta \\gamma^{2} h^{4} y_{1}^{2} y_{2}^{2} - 10 \\alpha \\beta h^{2} y_{3}^{2} - 10 \\alpha \\gamma h^{2} y_{2}^{2} + \\beta^{2} \\gamma^{2} h^{4} y_{1}^{4} - 10 \\beta \\gamma h^{2} y_{1}^{2} + 120\\right)}{120} & \\frac{\\beta y_{1} y_{3} \\left(\\alpha^{2} \\beta^{2} h^{4} y_{3}^{4} + 4 \\alpha^{2} \\beta \\gamma h^{4} y_{2}^{2} y_{3}^{2} + \\alpha^{2} \\gamma^{2} h^{4} y_{2}^{4} + 4 \\alpha \\beta^{2} \\gamma h^{4} y_{1}^{2} y_{3}^{2} + 4 \\alpha \\beta \\gamma^{2} h^{4} y_{1}^{2} y_{2}^{2} - 10 \\alpha \\beta h^{2} y_{3}^{2} - 10 \\alpha \\gamma h^{2} y_{2}^{2} + \\beta^{2} \\gamma^{2} h^{4} y_{1}^{4} - 10 \\beta \\gamma h^{2} y_{1}^{2} + 120\\right)}{120} & \\frac{\\gamma y_{1} y_{2} \\left(\\alpha^{2} \\beta^{2} h^{4} y_{3}^{4} + 4 \\alpha^{2} \\beta \\gamma h^{4} y_{2}^{2} y_{3}^{2} + \\alpha^{2} \\gamma^{2} h^{4} y_{2}^{4} + 4 \\alpha \\beta^{2} \\gamma h^{4} y_{1}^{2} y_{3}^{2} + 4 \\alpha \\beta \\gamma^{2} h^{4} y_{1}^{2} y_{2}^{2} - 10 \\alpha \\beta h^{2} y_{3}^{2} - 10 \\alpha \\gamma h^{2} y_{2}^{2} + \\beta^{2} \\gamma^{2} h^{4} y_{1}^{4} - 10 \\beta \\gamma h^{2} y_{1}^{2} + 120\\right)}{120}\\end{matrix}\\right]$"
],
"text/plain": [
"[alpha*y2*y3*(alpha**2*beta**2*h**4*y3**4 + 4*alpha**2*beta*gamma*h**4*y2**2*y3**2 + alpha**2*gamma**2*h**4*y2**4 + 4*alpha*beta**2*gamma*h**4*y1**2*y3**2 + 4*alpha*beta*gamma**2*h**4*y1**2*y2**2 - 10*alpha*beta*h**2*y3**2 - 10*alpha*gamma*h**2*y2**2 + beta**2*gamma**2*h**4*y1**4 - 10*beta*gamma*h**2*y1**2 + 120)/120, beta*y1*y3*(alpha**2*beta**2*h**4*y3**4 + 4*alpha**2*beta*gamma*h**4*y2**2*y3**2 + alpha**2*gamma**2*h**4*y2**4 + 4*alpha*beta**2*gamma*h**4*y1**2*y3**2 + 4*alpha*beta*gamma**2*h**4*y1**2*y2**2 - 10*alpha*beta*h**2*y3**2 - 10*alpha*gamma*h**2*y2**2 + beta**2*gamma**2*h**4*y1**4 - 10*beta*gamma*h**2*y1**2 + 120)/120, gamma*y1*y2*(alpha**2*beta**2*h**4*y3**4 + 4*alpha**2*beta*gamma*h**4*y2**2*y3**2 + alpha**2*gamma**2*h**4*y2**4 + 4*alpha*beta**2*gamma*h**4*y1**2*y3**2 + 4*alpha*beta*gamma**2*h**4*y1**2*y2**2 - 10*alpha*beta*h**2*y3**2 - 10*alpha*gamma*h**2*y2**2 + beta**2*gamma**2*h**4*y1**4 - 10*beta*gamma*h**2*y1**2 + 120)/120]"
]
},
"execution_count": 76,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"series = bs.modifying_integrator(u, f, A, b, order=5)\n",
"simplify(series)"
]
},
{
"cell_type": "markdown",
"id": "9f8e920f",
"metadata": {},
"source": [
"We can verify that the results match Eqn. (12) of CHV2007:"
]
},
{
"cell_type": "code",
"execution_count": 82,
"id": "0f8597c9",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle - \\frac{\\alpha \\beta y_{3}^{2}}{12} - \\frac{\\alpha \\gamma y_{2}^{2}}{12} - \\frac{\\beta \\gamma y_{1}^{2}}{12}$"
],
"text/plain": [
"-alpha*beta*y3**2/12 - alpha*gamma*y2**2/12 - beta*gamma*y1**2/12"
]
},
"execution_count": 82,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"s3 = sympy.Poly((series[0]-f[0])/(f[0]),h).coeffs()[1]\n",
"s3"
]
},
{
"cell_type": "code",
"execution_count": 84,
"id": "d1e59c6c",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\frac{\\alpha \\beta \\gamma \\left(\\alpha y_{2}^{2} y_{3}^{2} + \\beta y_{1}^{2} y_{3}^{2} + \\gamma y_{1}^{2} y_{2}^{2}\\right)}{60}$"
],
"text/plain": [
"alpha*beta*gamma*(alpha*y2**2*y3**2 + beta*y1**2*y3**2 + gamma*y1**2*y2**2)/60"
]
},
"execution_count": 84,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"s5 = sympy.Poly((series[0]-f[0])/(f[0]),h).coeffs()[0]\n",
"simplify(s5 - 6*one/5*s3**2)"
]
},
{
"cell_type": "markdown",
"id": "aea9616d",
"metadata": {},
"source": [
"# Lotka-Volterra"
]
},
{
"cell_type": "markdown",
"id": "db1aa2cb",
"metadata": {},
"source": [
"Next we consider using the explicit Euler method to solve the Lotka-Volterra model:\n",
"\n",
"$$\n",
" p'(t) = (2-q)p \\quad \\quad q'(t)=(p-1)q.\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 155,
"id": "b58d8ab7",
"metadata": {},
"outputs": [],
"source": [
"p, q = symbols('p,q')\n",
"u = [p,q]\n",
"f = np.array([p*(2-q),q*(p-1)])\n",
"\n",
"FE1 = rk.loadRKM('FE')\n",
"\n",
"dt = 0.35\n",
"T = 15.\n",
"IC = [1.5,2.25]\n",
"\n",
"f_ = lambdify([p,q],f)\n",
"\n",
"def f_vec(t,u):\n",
" return f_(*u)\n",
"\n",
"myivp = ivp.IVP(f=f_vec,u0=np.array(IC),T=T)\n",
"t0, y0 = FE1(myivp,dt=dt)\n",
"y0 = np.array(y0)\n",
"\n",
"f_ex = lambdify([p,q],f)\n",
"\n",
"def f_vec(t,u):\n",
" return f_ex(*u)\n",
"\n",
"myivp = ivp.IVP(f=f_vec,u0=np.array(IC),T=T)\n",
"BS5 = rk.loadRKM('BS5')\n",
"\n",
"t, y = BS5(myivp,errtol=1.e-10,dt=1.e-3)\n",
"y_exact = np.array(y)"
]
},
{
"cell_type": "markdown",
"id": "25945117",
"metadata": {},
"source": [
"The exact solution of this problem is periodic, but Euler's method produces an unstable trajectory. Here we use an especially large timestep in order to more clearly illustrate what will follow, but the qualitative behavior is the same for any step size."
]
},
{
"cell_type": "code",
"execution_count": 200,
"id": "19334a37",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 648x432 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"plt.figure(figsize=(9,6))\n",
"plt.plot(y_exact[:,1],y_exact[:,0],'-k',lw=2)\n",
"plt.plot(y0[:,1],y0[:,0],'--b')\n",
"plt.xlim(0,4)\n",
"plt.ylim(0,2.5)\n",
"plt.legend(['Exact solution','Explicit Euler, dt=0.1'],fontsize=15);"
]
},
{
"cell_type": "markdown",
"id": "ca24ef33",
"metadata": {},
"source": [
"Now we will derive a \"modifying integrator\". What this means is that we will determine a perturbed RHS such that when Euler's method is applied to the perturbed RHS, the result is the exact solution to the original Lotka-Volterra system. The perturbed system takes the form of a power series in $h$, and in order to compute with it we will truncate it at a certain order. We can compare the accuracy (and qualitative behavior) obtained by truncating at different orders."
]
},
{
"cell_type": "code",
"execution_count": 201,
"id": "b3c499d1",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}\\frac{p \\left(- h^{3} p^{3} q + 11 h^{3} p^{2} q^{2} - 11 h^{3} p^{2} q - 11 h^{3} p q^{3} + 26 h^{3} p q^{2} - 25 h^{3} p q + h^{3} q^{4} - 2 h^{3} q^{3} + 7 h^{3} q^{2} - 15 h^{3} q + 16 h^{3} - 4 h^{2} p^{2} q + 16 h^{2} p q^{2} - 24 h^{2} p q - 4 h^{2} q^{3} + 12 h^{2} q^{2} - 28 h^{2} q + 32 h^{2} - 12 h p q + 12 h q^{2} - 36 h q + 48 h - 24 q + 48\\right)}{24} & \\frac{q \\left(h^{3} p^{4} - 11 h^{3} p^{3} q + 8 h^{3} p^{3} + 11 h^{3} p^{2} q^{2} - 14 h^{3} p^{2} q + 10 h^{3} p^{2} - h^{3} p q^{3} - h^{3} p q^{2} - h^{3} p q + h^{3} + 4 h^{2} p^{3} - 16 h^{2} p^{2} q + 12 h^{2} p^{2} + 4 h^{2} p q^{2} + 4 h^{2} p - 4 h^{2} + 12 h p^{2} - 12 h p q + 12 h + 24 p - 24\\right)}{24}\\end{matrix}\\right]$"
],
"text/plain": [
"[p*(-h**3*p**3*q + 11*h**3*p**2*q**2 - 11*h**3*p**2*q - 11*h**3*p*q**3 + 26*h**3*p*q**2 - 25*h**3*p*q + h**3*q**4 - 2*h**3*q**3 + 7*h**3*q**2 - 15*h**3*q + 16*h**3 - 4*h**2*p**2*q + 16*h**2*p*q**2 - 24*h**2*p*q - 4*h**2*q**3 + 12*h**2*q**2 - 28*h**2*q + 32*h**2 - 12*h*p*q + 12*h*q**2 - 36*h*q + 48*h - 24*q + 48)/24, q*(h**3*p**4 - 11*h**3*p**3*q + 8*h**3*p**3 + 11*h**3*p**2*q**2 - 14*h**3*p**2*q + 10*h**3*p**2 - h**3*p*q**3 - h**3*p*q**2 - h**3*p*q + h**3 + 4*h**2*p**3 - 16*h**2*p**2*q + 12*h**2*p**2 + 4*h**2*p*q**2 + 4*h**2*p - 4*h**2 + 12*h*p**2 - 12*h*p*q + 12*h + 24*p - 24)/24]"
]
},
"execution_count": 201,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = FE1.A\n",
"b = FE1.b\n",
"max_order = 4\n",
"series = bs.modifying_integrator(u, f, A, b, order=max_order)\n",
"simplify(series)"
]
},
{
"cell_type": "code",
"execution_count": 202,
"id": "0db01f1f",
"metadata": {},
"outputs": [],
"source": [
"ymod = []\n",
"for order in range(2,max_order+1):\n",
" fs = simplify(np.array([term.series(h,0,order).removeO() for term in series]))\n",
" f_ = lambdify([p,q,h],fs)\n",
"\n",
" def f_p_vec(t,u,h=dt):\n",
" return f_(*u,h)\n",
"\n",
" myivp = ivp.IVP(f=f_p_vec,u0=np.array(IC),T=T)\n",
" _, y1 = FE1(myivp,dt=dt)\n",
" ymod.append(np.array(y1))"
]
},
{
"cell_type": "code",
"execution_count": 203,
"id": "50a688c5",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 648x432 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"colors = 'rgc'\n",
"plt.figure(figsize=(9,6))\n",
"plt.plot(y_exact[:,1],y_exact[:,0],'-k',lw=2)\n",
"plt.plot(y0[:,1],y0[:,0],'--b')\n",
"for j in range(max_order-1):\n",
" plt.plot(ymod[j][:,1],ymod[j][:,0],'--'+colors[j],lw=3,alpha=0.5)\n",
"plt.xlim(0,4)\n",
"plt.ylim(0,2.5)\n",
"plt.legend(['Exact solution','Explicit Euler, dt='+str(dt),\n",
" 'EE with modified flow to O(h)',\n",
" 'EE with modified flow to $O(h^2)$',\n",
" 'EE with modified flow to $O(h^3)$'],fontsize=15);"
]
},
{
"cell_type": "markdown",
"id": "4054d4ad",
"metadata": {},
"source": [
"We see that if we include one additional term, the resulting trajectory still grows, while with two additional terms the solution appears to be dissipative. With each additional term the solution gets closer to the exact solution of the original problem, and with three added terms it is hard to see the difference between them at this scale."
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.9.5"
}
},
"nbformat": 4,
"nbformat_minor": 5
}
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