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11: Solve some TODOs, do some TeX

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Koen Vendrig
2022-03-18 09:39:48 +01:00
committed by Kees van Kempen
parent f7ac2c3ff7
commit 1d7dec786c
1 changed files with 55 additions and 16 deletions
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Week 6/11 Parabolic PDEs: 1D Schrödinger Equation with Potential.ipynb Normal file → Executable file
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@ -160,12 +160,13 @@
"\n",
"with $\\delta_{ij} = 1 \\iff i = j$, and using the approximations $\\hbar = 1$, $m = 0.5$.\n",
"\n",
"\n",
"finite differences for second derivative to get a tridiagonal matrix (one lower and one higher than diagonal)\n",
"then plot eigenvalues\n",
"no time component -> not complex?\n",
"energy should be real\n",
"eigenvectors squared should be normalized"
"$$\n",
"%finite differences for second derivative to get a tridiagonal matrix (one lower and one higher than diagonal)\n",
"%then plot eigenvalues\n",
"%no time component -> not complex?\n",
"%energy should be real\n",
"%eigenvectors squared should be normalized\n",
"$$"
]
},
{
@ -201,7 +202,6 @@
" A tuple of the eigenvalues E and eigenfunctions \\Psi of the hamiltonian H.\n",
" \"\"\"\n",
" \n",
" # TODO: Is n the dimensionality of Psi?\n",
" n = len(x)\n",
" h = x[1] - x[0]\n",
" \n",
@ -286,7 +286,6 @@
"fig = plt.figure(figsize=(12,12))\n",
"\n",
"for n in range(the_first_few):\n",
" # TODO: Should SEQStat return normalized eigenfunctions?\n",
" # Normalize the found eigenfunctions using Riemann midpoint sums\n",
" prob_mass = np.sum((x[1:] - x[:-1])*(Psi[1:, n]**2 + Psi[:-1, n]**2)/2)\n",
" Psi[:, n] /= np.sqrt(prob_mass)\n",
@ -383,7 +382,50 @@
}
},
"source": [
"YOUR ANSWER HERE"
"To do the derivation of the Crank-Nicolson approach following $\\textit{Numerical Analysis}$ by Burden and Faires, we start from averaging our 'Forward-Difference method' at the jth step in t:\n",
"\n",
"$$\n",
"\\frac{w_{i,j+1}-w_{i,j}}{k}-\\alpha^2\\frac{w_{i+1,j}-2w_{i,j}+w_{i-1,j}}{h^2} = 0.\n",
"$$\n",
"\n",
"The local truncation error is:\n",
"\n",
"$$\n",
"\\tau_F = \\frac{k}{2}\\frac{\\partial^2 u}{\\partial t^2}(x_i,\\mu_j)+O(h^2).\n",
"$$\n",
"\n",
"We do the same for the 'Backward-Difference method' to get:\n",
"\n",
"$$\n",
"\\frac{w_{i,j+1}-w_{i,j}}{k}-\\alpha^2\\frac{w_{i+1,j+1}-2w_{i,j+1}+w_{i-1,j+1}}{h^2} = 0,\n",
"$$\n",
"\n",
"again with a local truncation error:\n",
"\n",
"$$\n",
"\\tau_B = -\\frac{k}{2}\\frac{\\partial^2 u}{\\partial t^2}(x_i,\\hat{u} _j)+O(h^2).\n",
"$$\n",
"\n",
"If we make the following assumption:\n",
"\n",
"$$\n",
"\\frac{\\partial^2 u}{\\partial t^2}(x_i,\\hat{\\mu}_j) \\approx \\frac{\\partial^2 u}{\\partial t^2}(x_i,\\mu_j),\n",
"$$\n",
"\n",
"Then we can get the 'averaged-difference method' to write:\n",
"\n",
"$$\n",
"\\frac{w_{i,j+1}-w_{i,j}}{k}-\\frac{\\alpha^2}{2}\\left[\\frac{w_{i+1,j}-2w_{i,j}+w_{i-1,j}}{h^2}+\\frac{w_{i+1,j+1}-2w_{i,j+1}+w_{i-1,j+1}}{h^2}\\right] = 0,\n",
"$$\n",
"\n",
"which now has an truncation error of the order $O(k^2+h^2)$. This is known as the Crank-Nicolson method and is represented in the form described above.\n",
"\n",
"For $\\lambda_1$ and $\\lambda_2$ we have the following expressions:\n",
"\n",
"$$\n",
"\\lambda_1 = \\alpha^2\\frac{k}{h^2}\\\\\n",
"\\lambda_2 = 1.\n",
"$$"
]
},
{
@ -432,8 +474,7 @@
"outputs": [],
"source": [
"def SEQDyn(x, t, V, f):\n",
" # YOUR CODE HERE\n",
" raise NotImplementedError()"
" # YOUR CODE HERE"
]
},
{
@ -457,8 +498,7 @@
"source": [
"# Animate your solution here ...\n",
"\n",
"# YOUR CODE HERE\n",
"raise NotImplementedError()"
"# YOUR CODE HERE"
]
},
{
@ -502,8 +542,7 @@
},
"outputs": [],
"source": [
"# YOUR CODE HERE\n",
"raise NotImplementedError()"
"# YOUR CODE HERE"
]
},
{
@ -567,7 +606,7 @@
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},