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inducerinducer
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Rewire DGF/Stationary sequencing
1 parent 89a0dc8 commit 7990c18

4 files changed

Lines changed: 252 additions & 54 deletions

‎cleared-demos/pdes/Discrete Greens Functions and Interaction Rank.ipynb‎

Lines changed: 67 additions & 8 deletions
Original file line numberDiff line numberDiff line change
@@ -33,7 +33,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"execution_count": 3,
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"id": "50cb54b8-95ae-4b38-a34b-3b68d9251ea5",
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"metadata": {},
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"outputs": [],
@@ -44,9 +44,17 @@
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"import matplotlib.pyplot as pt"
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]
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},
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{
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"cell_type": "markdown",
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"id": "d7ccce38-d1e2-483f-b811-ca580f36317e",
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"metadata": {},
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"source": [
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"Let's solve $u''=-30x^2$ with $u(0)=1$ and $u(1)=-1$."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"execution_count": 5,
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"id": "1f92e1b9-6975-458b-861c-64a7044e0c07",
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"metadata": {},
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"outputs": [],
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"h = mesh[1] - mesh[0]"
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]
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},
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{
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"cell_type": "markdown",
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"id": "972fef0b-fdf2-4709-b7bb-151573e1f583",
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"metadata": {},
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"source": [
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"Set up the system matrix `A` to carry out centered finite differences\n",
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"\n",
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"$$\n",
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"u''(x)\\approx \\frac{u(x+h) - 2u(x) + u(x-h)}{h^2}.\n",
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"$$\n",
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"\n",
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"Use `np.eye(n, k=...)`. What needs to be in the first and last row?"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"execution_count": 6,
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"id": "0865715a-b5c7-4cd6-9a85-ce751c158d29",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "markdown",
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"id": "80bf4822-86e1-4948-b801-43ea6f5598f2",
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"metadata": {},
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"source": [
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"Next, fix the right hand side: "
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"id": "a246c2c3-79ac-4966-9d46-34b6493eb16d",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "markdown",
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"id": "de5fe2de-537e-42a3-9415-4998a11699d6",
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"metadata": {},
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"source": [
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"Compute a reference solution `x_true` to the linear system:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"id": "de331c55-d62b-4edc-91de-79181e35342b",
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"metadata": {},
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"outputs": [],
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"source": [
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"x_true = la.solve(A, b)\n",
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"pt.plot(mesh, x_true)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "2d3e09f9-f508-4752-b3f6-4847d3fd3ad8",
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"metadata": {},
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"source": [
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"A = (np.eye(n, k=1) + -2*np.eye(n) + np.eye(n, k=-1))/h**2\n",
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"A[0] = 0\n",
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"A[-1] = 0\n",
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"A[0,0] = 1\n",
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"A[-1,-1] = 1"
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"Next, let's consider the influence of each individual RHS component:"
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]
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},
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{

‎cleared-demos/pdes/Stationary Iterative Methods.ipynb‎

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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"execution_count": 3,
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"metadata": {
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"collapsed": false,
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"jupyter": {
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"execution_count": 4,
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"metadata": {
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"collapsed": false,
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"jupyter": {
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}
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},
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"outputs": [],
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"source": []
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"source": [
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"A = (np.eye(n, k=1) + -2*np.eye(n) + np.eye(n, k=-1))/h**2\n",
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"A[0] = 0\n",
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"A[-1] = 0\n",
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"A[0,0] = 1\n",
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"A[-1,-1] = 1"
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]
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},
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{
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"cell_type": "markdown",
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"collapsed": false,
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"jupyter": {
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"collapsed": false,
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"jupyter": {
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {
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"collapsed": false,
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"jupyter": {
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},
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"cell_type": "code",
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"execution_count": 42,
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"execution_count": 9,
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"metadata": {
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"collapsed": false,
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"jupyter": {
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},
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"cell_type": "code",
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"execution_count": 10,
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"metadata": {
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"collapsed": false,
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"jupyter": {
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"cell_type": "code",
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"execution_count": 12,
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"collapsed": false,
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"jupyter": {
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},
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"cell_type": "code",
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"execution_count": 56,
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"collapsed": false,
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"jupyter": {
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},
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"cell_type": "code",
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"execution_count": 92,
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"execution_count": 174,
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"metadata": {
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"collapsed": false,
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"jupyter": {
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},
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"cell_type": "code",
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"execution_count": 240,
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"jupyter": {
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},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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},
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"cell_type": "code",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python3"
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},
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.8.3"
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"version": "3.14.3"
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}
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},
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"nbformat": 4,

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