{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "# import various packages\n",
    "\n",
    "import numpy as np                    # matrices & etc\n",
    "from matplotlib import pyplot as plt  # general plotting \n",
    "from matplotlib import patches        # plotting shapes\n",
    "from datetime import date             # date and time arithmetic\n",
    "from math import exp, pi, sin, cos, tan, sqrt, floor, log10, nan   # math functions\n",
    "import scipy.linalg as la             # linear algebra functions\n",
    "import os                             # operating system\n",
    "import matplotlib.cm as cm\n",
    "from matplotlib.colors import ListedColormap\n",
    "import scipy.signal as sg\n",
    "import scipy.stats as st\n",
    "import scipy.sparse.linalg as las\n",
    "from scipy import sparse\n",
    "import matplotlib\n",
    "from matplotlib import animation, rc\n",
    "from IPython.display import HTML\n",
    "# import ffmpeg\n",
    "\n",
    "# conda install -c conda-forge ffmpeg\n",
    "\n",
    "# function to make a numpy N-by-1 column vector\n",
    "# c=eda_cvec(v1, v2, ...) from a list of several\n",
    "# array-like entities v1, v2, including a number\n",
    "# a list of numbers, a tuple of numbers, an N-by-0 np array\n",
    "# and a N-by-1 np array. The function\n",
    "# also insures that, if v is an np array, that\n",
    "# c is a copy, as contrasted to a view, of it\n",
    "# It promotes integers to floats, and integers\n",
    "# and floats to complex, by context.\n",
    "# This version concatenates many argments,\n",
    "# whereas c=eda_cvec1(v1) takes just one argiment.\n",
    "# I recommend always using eda_cvec(v1, v2, ...)\n",
    "def gda_cvec(*argv):\n",
    "    t = int;\n",
    "    Nt = 0;\n",
    "    for a in argv:\n",
    "        v = gda_cvec1(a);\n",
    "        N,M = np.shape(v);\n",
    "        Nt = Nt + N;\n",
    "        if( N==0 ):\n",
    "            continue; # skip vector of zero length\n",
    "        if (t==int) and isinstance(v[0,0],float):\n",
    "            t=float;\n",
    "        elif isinstance(v[0,0],complex):\n",
    "            t=complex;\n",
    "    w = np.zeros((Nt,1),dtype=t);\n",
    "    Nt = 0;\n",
    "    for a in argv:\n",
    "        v = gda_cvec1(a);\n",
    "        N,M = np.shape(v);\n",
    "        w[Nt:Nt+N,0:1] = v;  # patch 20230418 was #w[Nt:Nt+N,0] = v[0:N,0];\n",
    "        Nt = Nt + N;\n",
    "    return w;\n",
    "\n",
    "# function to make a numpy N-by-1 column vector\n",
    "# c=gda_cvec1(v) from entity v that is array-like,\n",
    "# including a number, a list of numbers, a tuple\n",
    "# of numbers, an N-by-0 np array and a N-by1 np array.\n",
    "# It promotes integers to floats, and integers\n",
    "# and floats to complex, by context. The function\n",
    "# also insures that, if v is an np array, that\n",
    "# c is a copy, as contrasted to a view, of it.\n",
    "# This version takes just one input argment.\n",
    "# whereas c=gda_cvec(v1,v2,...) concatenates\n",
    "# many argiments.\n",
    "def gda_cvec1(v):\n",
    "    if isinstance(v, int) or isinstance(v, np.int32):\n",
    "        w = np.zeros((1,1),dtype=int);\n",
    "        w[0,0] = v;\n",
    "        return w;\n",
    "    elif isinstance(v, float):\n",
    "        w = np.zeros((1,1),dtype=float);\n",
    "        w[0,0] = v;\n",
    "        return w;\n",
    "    elif isinstance(v, complex):\n",
    "        w = np.zeros((1,1),dtype=complex);\n",
    "        w[0,0] = v;\n",
    "        return w;\n",
    "    elif isinstance(v, np.ndarray):\n",
    "        s = np.shape(v);\n",
    "        if len(s) == 1:\n",
    "            return np.copy(np.reshape(v,(s[0],1)));\n",
    "        else:\n",
    "            [r,c]=s;\n",
    "            if( c==1 ):\n",
    "                return(np.copy(v));\n",
    "            elif(r==1):\n",
    "                return(np.copy(v.T));\n",
    "            else:\n",
    "                raise TypeError(\"gda_cvec: %d by %d ndarray not allowed\" % (r, c));\n",
    "    elif isinstance(v, list):\n",
    "        r = len(v);\n",
    "        t = int;\n",
    "        for vi in v:\n",
    "            if isinstance(vi,int) or isinstance(vi, np.int32): #patch v->vi 20230418\n",
    "                pass;\n",
    "            elif isinstance(vi,float):\n",
    "                t=float;\n",
    "            elif isinstance(vi,complex):\n",
    "                t=complex;\n",
    "                break;\n",
    "            else:\n",
    "                raise TypeError(\"gda_cvec: list contains unsupported type %s\" % type(vi));\n",
    "        w = np.zeros((r,1),dtype=t);\n",
    "        w[:,0] = np.array(v); # patch v -> np.array(v)\n",
    "        return w;\n",
    "    elif isinstance(v, tuple):\n",
    "        r = len(v);\n",
    "        t = int;\n",
    "        for vi in v:\n",
    "            if isinstance(vi,int) or isinstance(vi, np.int32): #patch v->vi 20230418\n",
    "                pass;\n",
    "            elif isinstance(vi,float):\n",
    "                t=float;\n",
    "            elif isinstance(vi,complex):\n",
    "                t=complex;\n",
    "                break;\n",
    "            else:\n",
    "                raise TypeError(\"gda_cvec: tuple contains unsupported type %s\" % type(vi));\n",
    "        w = np.zeros((r,1),dtype=t);\n",
    "        w[:,0] = np.array(list(v)); # patch v -> np.array(list(v));\n",
    "        return w;\n",
    "    else:\n",
    "        raise TypeError(\"gda_cvec: %s not supported\" % type(v));\n",
    "\n",
    "        \n",
    "# gda_draw function makes a \"pictorial matrix equation\"\n",
    "# arguments are vectors, matrices and strings\n",
    "# which are plotted in the order that the appear\n",
    "# except that strings starting with 'title ' are plotted\n",
    "# under the subseqeunt matrix or vector\n",
    "# always returns a status of 1\n",
    "def gda_draw(*argv):\n",
    "    DOCOLOR=True;\n",
    "    if( DOCOLOR ):\n",
    "        bwcmap = matplotlib.colormaps['jet'];\n",
    "    else:\n",
    "        bw = np.zeros((256,4));\n",
    "        v = 0.9*(256 - np.linspace( 0, 255, 256 ))/255;\n",
    "        bw[:,0] = v;\n",
    "        bw[:,1] = v;\n",
    "        bw[:,2] = v;\n",
    "        bw[:,3] = np.ones(256);\n",
    "        bwcmap = ListedColormap(bw);\n",
    "    # size of plot\n",
    "    W = 16;\n",
    "    H = 4;\n",
    "    fig1 = plt.figure(1);\n",
    "    # figsize width and height in inches\n",
    "    fig1.set_size_inches(W,H);\n",
    "    ax1 = plt.subplot(1,1,1);\n",
    "    plt.axis([0, W, -H/2, H/2]);\n",
    "    plt.axis('off');\n",
    "    LM = W/6;    # matrix width and heoght\n",
    "    LV = W/40;   # vector width\n",
    "    FS = 0.12;    # character width\n",
    "    TO = 0.4;    # title vertical offset\n",
    "    SP = 0.2;    # space between objects\n",
    "    LS = 0.2;    # leading space\n",
    "    p = LS; # starting x-position\n",
    "    istitle=0; # flags presence of a title\n",
    "    for a in argv:\n",
    "        if isinstance(a,np.ndarray):\n",
    "            sh = np.shape(a);\n",
    "            if len(sh) == 1:  # conversion to nx1 array\n",
    "                n = sh[0];\n",
    "                m = 1;\n",
    "                ap = a;\n",
    "                a = np.zeros((n,1));\n",
    "                a[:,0] = ap;\n",
    "            else:\n",
    "                n = sh[0];\n",
    "                m = sh[1];\n",
    "            if m==1:\n",
    "                pold=p;\n",
    "                left=p;\n",
    "                right=p+LV;\n",
    "                bottom=-LM/2;\n",
    "                top=LM/2;\n",
    "                plt.imshow( a, cmap=bwcmap, vmin=np.min(a), vmax=np.max(a), extent=(left,right,bottom,top) );\n",
    "                p = p+LV;\n",
    "                pm = (p+pold)/2;\n",
    "                if istitle:\n",
    "                    plt.text(pm,-(LM/2)-TO,titlestr,horizontalalignment='center');\n",
    "                    istitle=0;\n",
    "                p = p+SP;\n",
    "            else:\n",
    "                pold=p;\n",
    "                left=p;\n",
    "                right=p+LM;\n",
    "                bottom=-LM/2;\n",
    "                top=LM/2;\n",
    "                plt.imshow( a, cmap=bwcmap, vmin=np.min(a), vmax=np.max(a), extent=(left,right,bottom,top) );\n",
    "                p = p+LM;\n",
    "                pm = (p+pold)/2;\n",
    "                if istitle:\n",
    "                    plt.text(pm,-(LM/2)-TO,titlestr,horizontalalignment='center');\n",
    "                    istitle=0;\n",
    "                p = p+SP;\n",
    "        elif isinstance(a,str):\n",
    "            ns = len(a);\n",
    "            istitle=0;\n",
    "            if( ns>=6 ):\n",
    "                if 'title ' in a[0:6]:\n",
    "                    istitle=1;\n",
    "                    titlestr=a[6:];\n",
    "            if( istitle != 1):\n",
    "                plt.text(p,0,a);\n",
    "                p = p + ns*FS + SP;\n",
    "    plt.show();\n",
    "    return 1;\n",
    "\n",
    "# bandpass filter, used in seismological example, but hand\n",
    "# in a variety of settings involving time series\n",
    "def gda_chebyshevfilt(d, Dt, flow, fhigh):\n",
    "    # (dout,u,v)=gda_chebyshevfilt(d, Dt, flow, fhigh);\n",
    "    # chebyshev IIR bandpass filter\n",
    "    # d - input array of data\n",
    "    # Dt - sampling interval\n",
    "    # flow - low pass frequency, Hz\n",
    "    # fhigh - high pass frequency, Hz\n",
    "    # dout - output array of data\n",
    "    # u - the numerator filter\n",
    "    # v - the denominator filter\n",
    "    # these filters can be used again using dout=filter(u,v,din);\n",
    "\n",
    "    # make sure input timeseries is a column vector\n",
    "    s = np.shape(d);\n",
    "    N = s[0];\n",
    "    if(N==1):\n",
    "        dd = np.zeros((N,1));\n",
    "        dd[:,0] = d;\n",
    "    else:\n",
    "        dd=d;\n",
    "        \n",
    "    # sampling rate\n",
    "    rate=1/Dt;\n",
    "\n",
    "    # ripple parameter, set to ten percent\n",
    "    ripple=0.1;  \n",
    "\n",
    "    # normalise frequency\n",
    "    fl=2.0*flow/rate;\n",
    "    fh=2.0*fhigh/rate;\n",
    "\n",
    "    # center frequency \n",
    "    cf = 4 * tan( (fl*pi/2) ) * tan( (fh*pi/2) );\n",
    "\n",
    "    # bandwidth\n",
    "    bw = 2 * ( tan( (fh*pi/2) ) - tan( (fl*pi/2) ) );\n",
    "\n",
    "    # ripple parameter factor\n",
    "    rpf = sqrt((sqrt((1.0+1.0/(ripple*ripple))) + 1.0/ripple));\n",
    "    a = 0.5*(rpf-1.0/rpf);\n",
    "    b = 0.5*(rpf+1.0/rpf);\n",
    "\n",
    "    u=np.zeros((5,1));\n",
    "    v=np.zeros((5,1));\n",
    "    theta = 3*pi/4;\n",
    "    sr = a * cos(theta);\n",
    "    si = b * sin(theta);\n",
    "    es = sqrt(sr*sr+si*si);\n",
    "    tmp= 16.0 - 16.0*bw*sr + 8.0*cf + 4.0*es*es*bw*bw - 4.0*bw*cf*sr + cf*cf;\n",
    "    v[0,0] = 1.0;\n",
    "    v[1,0] = 4.0*(-16.0 + 8.0*bw*sr - 2.0*bw*cf*sr + cf*cf)/tmp;\n",
    "    v[2,0] = (96.0 - 16.0*cf - 8.0*es*es*bw*bw + 6.0*cf*cf)/tmp;\n",
    "    v[3,0] = (-64.0 - 32.0*bw*sr + 8.0*bw*cf*sr + 4.0*cf*cf)/tmp;\n",
    "    v[4,0] = (16.0 + 16.0*bw*sr + 8.0*cf + 4.0*es*es*bw*bw + 4.0*bw*cf*sr + cf*cf)/tmp;\n",
    "    tmp = 4.0*es*es*bw*bw/tmp;\n",
    "    u[0,0] = tmp;\n",
    "    u[1,0] = 0.0;\n",
    "    u[2,0] = -2.0*tmp;\n",
    "    u[3,0] = 0.0;\n",
    "    u[4,0] = tmp;\n",
    "\n",
    "    dout = sg.lfilter(u.ravel(),v.ravel(),dd.ravel());\n",
    "    return (gda_cvec(dout),gda_cvec(u),gda_cvec(v));\n",
    "\n",
    "# function to perform the multiplication FT (F v)\n",
    "def gdaFTFmul(v):\n",
    "    # this function is used by bicongugate gradient to\n",
    "    # solve the generalized least squares problem Fm=F.\n",
    "    # Note that \"F\" must be called \"gdaFsparse\".\n",
    "    global gdaFsparse;\n",
    "    s = np.shape(v);\n",
    "    if(len(s)==1):\n",
    "        vv = np.zeros((s[0],1));\n",
    "        vv[:,0] = v;\n",
    "    else:\n",
    "        vv=v;\n",
    "    # weirdly, \"*\" multiplies sparse matrices just fine\n",
    "    temp = gdaFsparse*vv;\n",
    "    return gdaFsparse.transpose()*temp;\n",
    "\n",
    "# function to perform the multiplication FT (G v)\n",
    "def gda_FTFmul(v):\n",
    "    # this function is used by bicongugate gradient to\n",
    "    # solve the generalized least squares problem Fm=F.\n",
    "    # Note that \"F\" must be called \"gdaFsparse\".\n",
    "    global gdaFsparse;\n",
    "    s = np.shape(v);\n",
    "    if(len(s)==1):\n",
    "        vv = np.zeros((s[0],1));\n",
    "        vv[:,0] = v;\n",
    "    else:\n",
    "        vv=v;\n",
    "    # weirdly, \"*\" multiplies sparse matrices just fine\n",
    "    temp = gdaFsparse*vv;\n",
    "    return gdaFsparse.transpose()*temp;\n",
    "\n",
    "\n",
    "# function to perform the multiplication FT f\n",
    "def gda_FTFrhs(f):\n",
    "    # this function is used by bicongugate gradient to\n",
    "    # solve the generalized least squares problem Fm=F.\n",
    "    # Note that \"F\" must be called \"gdaFsparse\".\n",
    "    global gdaFsparse;\n",
    "    return gdaFsparse.transpose()*f;"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Total number of equatons is 19200\n",
      "Interior equations:\n",
      "    18720 rows, expected 18252\n",
      "    112320 elements, expected 112320\n",
      "Bottom Boundary Condition\n",
      "    240 rows, expected 240\n",
      "    320 elements, expected 320\n",
      "Top Boundary Condition\n",
      "    240 rows, expected 240\n",
      "    400 elements, expected 400\n",
      "19200 rows, 19200 expected\n",
      "iter:  1 max(abs(s-Dm)) / max(abs(s)): 7.913822e-11\n"
     ]
    },
    {
     "data": {
      "image/png": "[encoded data, 74880 characters]",
      "text/plain": [
       "<Figure size 1600x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter:  50 max(abs(s-Dm)) / max(abs(s)): 7.974920e-11\n"
     ]
    },
    {
     "data": {
      "image/png": "[encoded data, 82252 characters]",
      "text/plain": [
       "<Figure size 1600x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter:  100 max(abs(s-Dm)) / max(abs(s)): 5.028039e-11\n"
     ]
    },
    {
     "data": {
      "image/png": "[encoded data, 90608 characters]",
      "text/plain": [
       "<Figure size 1600x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter:  150 max(abs(s-Dm)) / max(abs(s)): 9.742988e-11\n"
     ]
    },
    {
     "data": {
      "image/png": "[encoded data, 98748 characters]",
      "text/plain": [
       "<Figure size 1600x400 with 1 Axes>"
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    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter:  200 max(abs(s-Dm)) / max(abs(s)): 5.662501e-11\n"
     ]
    },
    {
     "data": {
      "image/png": "[encoded data, 105612 characters]",
      "text/plain": [
       "<Figure size 1600x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter:  250 max(abs(s-Dm)) / max(abs(s)): 4.659514e-11\n"
     ]
    },
    {
     "data": {
      "image/png": "[encoded data, 106860 characters]",
      "text/plain": [
       "<Figure size 1600x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter:  300 max(abs(s-Dm)) / max(abs(s)): 1.032457e-10\n"
     ]
    },
    {
     "data": {
      "image/png": "[encoded data, 108560 characters]",
      "text/plain": [
       "<Figure size 1600x400 with 1 Axes>"
      ]
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     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter:  350 max(abs(s-Dm)) / max(abs(s)): 6.060903e-11\n"
     ]
    },
    {
     "data": {
      "image/png": "[encoded data, 110556 characters]",
      "text/plain": [
       "<Figure size 1600x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter:  400 max(abs(s-Dm)) / max(abs(s)): 7.530245e-11\n"
     ]
    },
    {
     "data": {
      "image/png": "[encoded data, 109708 characters]",
      "text/plain": [
       "<Figure size 1600x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter:  450 max(abs(s-Dm)) / max(abs(s)): 4.530516e-11\n"
     ]
    },
    {
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      "text/plain": [
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     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter:  500 max(abs(s-Dm)) / max(abs(s)): 3.924753e-11\n"
     ]
    },
    {
     "data": {
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      "text/plain": [
       "<Figure size 1600x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter:  550 max(abs(s-Dm)) / max(abs(s)): 1.578198e-10\n"
     ]
    },
    {
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      "text/plain": [
       "<Figure size 1600x400 with 1 Axes>"
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     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter:  600 max(abs(s-Dm)) / max(abs(s)): 8.668862e-11\n"
     ]
    },
    {
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      "text/plain": [
       "<Figure size 1600x400 with 1 Axes>"
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    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter:  650 max(abs(s-Dm)) / max(abs(s)): 3.760049e-11\n"
     ]
    },
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    {
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     "text": [
      "iter:  700 max(abs(s-Dm)) / max(abs(s)): 5.560079e-11\n"
     ]
    },
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     "text": [
      "iter:  750 max(abs(s-Dm)) / max(abs(s)): 4.243647e-11\n"
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     "text": [
      "iter:  800 max(abs(s-Dm)) / max(abs(s)): 5.804912e-11\n"
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     "text": [
      "iter:  850 max(abs(s-Dm)) / max(abs(s)): 6.240535e-11\n"
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     "text": [
      "iter:  900 max(abs(s-Dm)) / max(abs(s)): 5.089159e-11\n"
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     "text": [
      "iter:  950 max(abs(s-Dm)) / max(abs(s)): 7.344913e-11\n"
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     "text": [
      "iter:  1000 max(abs(s-Dm)) / max(abs(s)): 3.717052e-11\n"
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     "text": [
      "iter:  1050 max(abs(s-Dm)) / max(abs(s)): 1.210943e-10\n"
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    },
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     "text": [
      "iter:  1100 max(abs(s-Dm)) / max(abs(s)): 3.981990e-11\n"
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     "text": [
      "iter:  1150 max(abs(s-Dm)) / max(abs(s)): 9.161021e-11\n"
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     "text": [
      "iter:  1200 max(abs(s-Dm)) / max(abs(s)): 3.674382e-11\n"
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    },
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     "text": [
      "iter:  1250 max(abs(s-Dm)) / max(abs(s)): 8.584301e-11\n"
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     "text": [
      "iter:  1300 max(abs(s-Dm)) / max(abs(s)): 6.917610e-11\n"
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     "text": [
      "iter:  1350 max(abs(s-Dm)) / max(abs(s)): 5.222432e-11\n"
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     "text": [
      "iter:  1400 max(abs(s-Dm)) / max(abs(s)): 6.840376e-11\n"
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    },
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     "output_type": "stream",
     "text": [
      "iter:  1450 max(abs(s-Dm)) / max(abs(s)): 4.313980e-11\n"
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    },
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     "text": [
      "iter:  1500 max(abs(s-Dm)) / max(abs(s)): 3.529401e-11\n"
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    },
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     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter:  1550 max(abs(s-Dm)) / max(abs(s)): 7.536888e-11\n"
     ]
    },
    {
     "data": {
      "image/png": "[encoded data, 100764 characters]",
      "text/plain": [
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     },
     "metadata": {},
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    {
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     "output_type": "stream",
     "text": [
      "iter:  1600 max(abs(s-Dm)) / max(abs(s)): 4.138929e-11\n"
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    },
    {
     "data": {
      "image/png": "[encoded data, 101332 characters]",
      "text/plain": [
       "<Figure size 1600x400 with 1 Axes>"
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     },
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    },
    {
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     "output_type": "stream",
     "text": [
      "iter:  1650 max(abs(s-Dm)) / max(abs(s)): 1.722899e-10\n"
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    },
    {
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     },
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter:  1700 max(abs(s-Dm)) / max(abs(s)): 6.315791e-11\n"
     ]
    },
    {
     "data": {
      "image/png": "[encoded data, 101664 characters]",
      "text/plain": [
       "<Figure size 1600x400 with 1 Axes>"
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     },
     "metadata": {},
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter:  1750 max(abs(s-Dm)) / max(abs(s)): 7.928280e-11\n"
     ]
    },
    {
     "data": {
      "image/png": "[encoded data, 101644 characters]",
      "text/plain": [
       "<Figure size 1600x400 with 1 Axes>"
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     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter:  1800 max(abs(s-Dm)) / max(abs(s)): 9.330599e-11\n"
     ]
    },
    {
     "data": {
      "image/png": "[encoded data, 102216 characters]",
      "text/plain": [
       "<Figure size 1600x400 with 1 Axes>"
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     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter:  1850 max(abs(s-Dm)) / max(abs(s)): 6.553809e-11\n"
     ]
    },
    {
     "data": {
      "image/png": "[encoded data, 102548 characters]",
      "text/plain": [
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     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter:  1900 max(abs(s-Dm)) / max(abs(s)): 1.027628e-10\n"
     ]
    },
    {
     "data": {
      "image/png": "[encoded data, 101948 characters]",
      "text/plain": [
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     "metadata": {},
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter:  1950 max(abs(s-Dm)) / max(abs(s)): 8.008094e-11\n"
     ]
    },
    {
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    {
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     "text": [
      "done with  2000  iterations\n"
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       "HWRhdGEAAAABAAAAAExhdmY1OC40NS4xMDA=\n",
       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "[encoded data, 26388 characters]",
      "text/plain": [
       "<Figure size 640x480 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# convection at high Rayleigh number\n",
    "\n",
    "Ra = 100000.0;\n",
    "Tbot = 1.0;\n",
    "DTpert = 0.15;\n",
    "NITER=2000;\n",
    "SNAPSHOT=50;\n",
    "\n",
    "# smoothing matrix\n",
    "W=1;\n",
    "if( W == 0 ):\n",
    "    w = np.array( [ [0.0, 0.0, 0.0],\n",
    "                    [0.0, 1.0, 0.0],\n",
    "                    [0.0, 0.0, 0.0] ] );\n",
    "elif( W == 1 ):\n",
    "    w = np.array( [ [0.1, 0.1, 0.1],\n",
    "                    [0.1, 0.2, 0.1],\n",
    "                    [0.1, 0.1, 0.1] ] );\n",
    "\n",
    "\n",
    "# x-axis\n",
    "Lx = 80;\n",
    "xmin = 0.0;\n",
    "xmax = 1.0;\n",
    "Dx = (xmax-xmin)/(Lx-1);\n",
    "x = gda_cvec(Dx*np.linspace(0,Lx-1,Lx));\n",
    "Dxr = 1.0/Dx;\n",
    "TDxr = 1.0/(2.0*Dx);\n",
    "Dxr2 = 1.0/(Dx**2);\n",
    "\n",
    "# z-axis\n",
    "Lz = 80; # must be even\n",
    "zmin=0.0;\n",
    "zmax=1.0;\n",
    "Dz = (zmax-zmin)/(Lz-1);\n",
    "z = gda_cvec(Dz*np.linspace(0,Lz-1,Lz));\n",
    "Dzr = 1.0/Dz;\n",
    "TDzr = 1.0/(2.0*Dz);\n",
    "Dzr2 = 1.0/(Dz**2);\n",
    "\n",
    "Lc = 3; # three components, q=0, ux=1, uz=2\n",
    "M = Lx*Lz*Lc; # number of unknowns\n",
    " \n",
    "# lookup tables translate between (i,j) image an linear k ordering\n",
    "iofk = np.zeros((M,1),dtype=int); # x position\n",
    "jofk = np.zeros((M,1),dtype=int); # y position\n",
    "cofk = np.zeros((M,1),dtype=int); # component q=0, ux=1, uz=2\n",
    "kofijc = np.zeros((Lx,Lz,Lc),dtype=int);\n",
    "k=0;\n",
    "for ic in range(Lc):\n",
    "    for ix in range(Lx):\n",
    "        for iz in range(Lz):\n",
    "            iofk[k,0]=ix;\n",
    "            jofk[k,0]=iz;\n",
    "            cofk[k,0]=ic;\n",
    "            kofijc[ix,iz,ic]=k;\n",
    "            k=k+1;\n",
    "            \n",
    "# two Gaussian temperature blobs of opposite sign\n",
    "sigmax = 0.1;\n",
    "sigmax2 = sigmax**2;\n",
    "sigmaz = 0.05;\n",
    "sigmaz2 = sigmaz**2;\n",
    "x0 = 1.0*xmax/3.0;\n",
    "z0 = zmax/2.0;\n",
    "Tx = np.exp(-0.5*np.power(x-x0,2)/sigmax2);\n",
    "Tz = np.exp(-0.5*np.power(z-z0,2)/sigmaz2);\n",
    "Ta = DTpert*np.matmul(Tx,Tz.T);\n",
    "x0 = 2.0*xmax/3.0;\n",
    "z0 = xmax/2.0;\n",
    "Tx = np.exp(-0.5*np.power(x-x0,2)/sigmax2);\n",
    "Tz = np.exp(-0.5*np.power(z-z0,2)/sigmaz2);\n",
    "Tb = -DTpert*np.matmul(Tx,Tz.T);\n",
    "T = np.matmul( np.ones((Lx,1)), (Tbot*(zmax-z)/zmax).T )+Ta+Tb;\n",
    "\n",
    "TUX=np.zeros((Lx,Lz,NITER));\n",
    "TUZ=np.zeros((Lx,Lz,NITER));\n",
    "TIMG = np.zeros((Lx,Lz,NITER));\n",
    "TIMG[0:Lx,0:Lz,0]=T[0:Lx,0:Lz];\n",
    "\n",
    "Pmax = 1000000;\n",
    "Dr = np.zeros((Pmax,1),dtype=int);\n",
    "Dc = np.zeros((Pmax,1),dtype=int);\n",
    "Dv = np.zeros((Pmax,1));\n",
    "\n",
    "nr = 0; # row counter\n",
    "ne = 0; # elememt counter\n",
    "# interior equations, of which there are three\n",
    "#                grad(q) - Del-squared u  = Ra T t\n",
    "#          1:    dq/dx - d2 ux/ dx2 - d2 ux/ dz2 = 0\n",
    "#          2:    dq/dz - d2 uz/ dx2 - d2 uz/ dz2 = Ra T\n",
    "#                div u = 0\n",
    "#          3:    d ux / dx + d uz / dz = 0\n",
    "for ix in range(Lx):\n",
    "    for iz in range(1,Lz-1):\n",
    "        # 3x3 grid of q's, that wrap at the left & right edges\n",
    "        if( ix==0 ):\n",
    "            k00q = kofijc[Lx-1,iz-1,0];\n",
    "            k01q = kofijc[Lx-1,iz,0];\n",
    "            k02q = kofijc[Lx-1,iz+1,0];\n",
    "        else:\n",
    "            k00q = kofijc[ix-1,iz-1,0];\n",
    "            k01q = kofijc[ix-1,iz,0];\n",
    "            k02q = kofijc[ix-1,iz+1,0];\n",
    "        k10q = kofijc[ix,iz-1,0];\n",
    "        k11q = kofijc[ix,iz,0];\n",
    "        k12q = kofijc[ix,iz+1,0];\n",
    "        if( ix==(Lx-1) ):\n",
    "            k20q = kofijc[0,iz-1,0];\n",
    "            k21q = kofijc[0,iz,0];\n",
    "            k22q = kofijc[0,iz+1,0];\n",
    "        else:\n",
    "            k20q = kofijc[ix+1,iz-1,0];\n",
    "            k21q = kofijc[ix+1,iz,0];\n",
    "            k22q = kofijc[ix+1,iz+1,0];\n",
    "        # 3x3 grid of ux's\n",
    "        if( ix==0 ):\n",
    "            k00x = kofijc[Lx-1,iz-1,1];\n",
    "            k01x = kofijc[Lx-1,iz,1];\n",
    "            k02x = kofijc[Lx-1,iz+1,1];\n",
    "        else:\n",
    "            k00x = kofijc[ix-1,iz-1,1];\n",
    "            k01x = kofijc[ix-1,iz,1];\n",
    "            k02x = kofijc[ix-1,iz+1,1];\n",
    "        k10x = kofijc[ix,iz-1,1];\n",
    "        k11x = kofijc[ix,iz,1];\n",
    "        k12x = kofijc[ix,iz+1,1];\n",
    "        if( ix==(Lx-1) ):\n",
    "            k20x = kofijc[0,iz-1,1];\n",
    "            k21x = kofijc[0,iz,1];\n",
    "            k22x = kofijc[0,iz+1,1];\n",
    "        else:\n",
    "            k20x = kofijc[ix+1,iz-1,1];\n",
    "            k21x = kofijc[ix+1,iz,1];\n",
    "            k22x = kofijc[ix+1,iz+1,1];\n",
    "        # 3x3 grid of uz's\n",
    "        if( ix==0 ):\n",
    "            k00z = kofijc[Lx-1,iz-1,2];\n",
    "            k01z = kofijc[Lx-1,iz,2];\n",
    "            k02z = kofijc[Lx-1,iz+1,2];\n",
    "        else:\n",
    "            k00z = kofijc[ix-1,iz-1,2];\n",
    "            k01z = kofijc[ix-1,iz,2];\n",
    "            k02z = kofijc[ix-1,iz+1,2];\n",
    "        k10z = kofijc[ix,iz-1,2];\n",
    "        k11z = kofijc[ix,iz,2];\n",
    "        k12z = kofijc[ix,iz+1,2];\n",
    "        if( ix==(Lx-1) ):\n",
    "            k20z = kofijc[0,iz-1,2];\n",
    "            k21z = kofijc[0,iz,2];\n",
    "            k22z = kofijc[0,iz+1,2];\n",
    "        else:\n",
    "            k20z = kofijc[ix+1,iz-1,2];\n",
    "            k21z = kofijc[ix+1,iz,2];\n",
    "            k22z = kofijc[ix+1,iz+1,2];\n",
    "        # 1: dq/dx - d2 ux/ dx2 - d2 ux/ dz2 = 0 (7 elements)\n",
    "        Dr[ne,0] = nr; # dq/dx\n",
    "        Dc[ne,0] = k01q;\n",
    "        Dv[ne,0] = -TDxr;\n",
    "        ne=ne+1;\n",
    "        Dr[ne,0] = nr;  \n",
    "        Dc[ne,0] = k21q;\n",
    "        Dv[ne,0] = TDxr;\n",
    "        ne=ne+1;\n",
    "        Dr[ne,0] = nr; # - d2 ux/ dx2 \n",
    "        Dc[ne,0] = k01x;\n",
    "        Dv[ne,0] = -Dxr2;\n",
    "        ne=ne+1;\n",
    "        Dr[ne,0] = nr;\n",
    "        Dc[ne,0] = k11x;\n",
    "        Dv[ne,0] = 2.0*Dxr2 + 2.0*Dzr2; # includes absorbed terms\n",
    "        ne=ne+1;\n",
    "        Dr[ne,0] = nr;\n",
    "        Dc[ne,0] = k21x;\n",
    "        Dv[ne,0] = -Dxr2;\n",
    "        ne=ne+1;\n",
    "        Dr[ne,0] = nr; # - d2 ux/ dz2 \n",
    "        Dc[ne,0] = k10x;\n",
    "        Dv[ne,0] = -Dzr2;\n",
    "        ne=ne+1;\n",
    "        # Dr[ne,0] = nr; Absorbed into previous instance\n",
    "        # Dc[ne,0] = k11x;\n",
    "        # Dv[ne,0] = 2.0*Dzr2;\n",
    "        # ne=ne+1;\n",
    "        Dr[ne,0] = nr;\n",
    "        Dc[ne,0] = k12x;\n",
    "        Dv[ne,0] = -Dzr2;\n",
    "        ne=ne+1;\n",
    "        nr=nr+1;\n",
    "        # 2: dq/dz - d2 uz/ dx2 - d2 uz/ dz2 = Ra T  (7 elements)\n",
    "        Dr[ne,0] = nr; # dq/dz\n",
    "        Dc[ne,0] = k10q;\n",
    "        Dv[ne,0] = -TDzr;\n",
    "        ne=ne+1;\n",
    "        Dr[ne,0] = nr;\n",
    "        Dc[ne,0] = k12q;\n",
    "        Dv[ne,0] = TDzr;\n",
    "        ne=ne+1;\n",
    "        Dr[ne,0] = nr; # - d2 uz/ dx2\n",
    "        Dc[ne,0] = k01z;\n",
    "        Dv[ne,0] = -Dxr2;\n",
    "        ne=ne+1;\n",
    "        Dr[ne,0] = nr;\n",
    "        Dc[ne,0] = k11z;\n",
    "        Dv[ne,0] = 2.0*Dxr2 + 2.0*Dzr2; # includes absorbed terms\n",
    "        ne=ne+1;\n",
    "        Dr[ne,0] = nr;\n",
    "        Dc[ne,0] = k21z;\n",
    "        Dv[ne,0] = -Dxr2;\n",
    "        ne=ne+1;\n",
    "        Dr[ne,0] = nr; # - d2 uz/ dz2\n",
    "        Dc[ne,0] = k10z;\n",
    "        Dv[ne,0] = -Dzr2;\n",
    "        ne=ne+1;\n",
    "        # Dr[ne,0] = nr; Absorbed into previous instance\n",
    "        # Dc[ne,0] = k11z;\n",
    "        # Dv[ne,0] = 2.0*Dzr2;\n",
    "        # ne=ne+1;\n",
    "        Dr[ne,0] = nr;\n",
    "        Dc[ne,0] = k12z;\n",
    "        Dv[ne,0] = -Dzr2;\n",
    "        ne=ne+1;\n",
    "        nr=nr+1;\n",
    "        # 3: d ux / dx + d uz / dz = 0  (4 elements)\n",
    "        Dr[ne,0] = nr;  # d ux / dx \n",
    "        Dc[ne,0] = k01x;\n",
    "        Dv[ne,0] = -TDxr;\n",
    "        ne=ne+1;\n",
    "        Dr[ne,0] = nr; \n",
    "        Dc[ne,0] = k21x;\n",
    "        Dv[ne,0] = TDxr;\n",
    "        ne=ne+1;\n",
    "        Dr[ne,0] = nr; # d uz / dz \n",
    "        Dc[ne,0] = k10z;\n",
    "        Dv[ne,0] = -TDzr;\n",
    "        ne=ne+1;\n",
    "        Dr[ne,0] = nr; \n",
    "        Dc[ne,0] = k12z;\n",
    "        Dv[ne,0] = TDzr;\n",
    "        ne=ne+1;\n",
    "        nr=nr+1;\n",
    "\n",
    "print(\"Total number of equatons is %d\" % (M) );\n",
    "print(\"Interior equations:\")\n",
    "nrtrue = 3*(Lx-2)*(Lz-2);\n",
    "print(\"    %d rows, expected %d\" % (nr, nrtrue ));\n",
    "netrue = 18*(Lx)*(Lz-2);\n",
    "print(\"    %d elements, expected %d\" % (ne, netrue ));\n",
    "\n",
    "nrstart=nr;\n",
    "nestart=ne;\n",
    "# bottom z=0 boundary, q = d ux / dz = uz = 0\n",
    "iz=0;\n",
    "for ix in range(Lx):\n",
    "    k11q = kofijc[ix,iz,0];\n",
    "    k12q = kofijc[ix,iz+1,0];\n",
    "    k11x = kofijc[ix,iz,1];\n",
    "    k12x = kofijc[ix,iz+1,2];\n",
    "    k11z = kofijc[ix,iz,2];\n",
    "    Dr[ne,0] = nr; # q = 0 (1 element2)\n",
    "    Dc[ne,0] = k11q;\n",
    "    Dv[ne,0] = 1.0;\n",
    "    ne=ne+1;\n",
    "    nr=nr+1;\n",
    "    Dr[ne,0] = nr; # d ux / dz = 0 (2 elements)\n",
    "    Dc[ne,0] = k11x;\n",
    "    Dv[ne,0] = -Dzr;\n",
    "    ne=ne+1;\n",
    "    Dr[ne,0] = nr;\n",
    "    Dc[ne,0] = k12x;\n",
    "    Dv[ne,0] = Dzr;\n",
    "    ne=ne+1;\n",
    "    nr=nr+1;\n",
    "    Dr[ne,0] = nr; # uz = 0 (1 element)\n",
    "    Dc[ne,0] = k11z;\n",
    "    Dv[ne,0] = 1.0;\n",
    "    ne=ne+1;\n",
    "    nr=nr+1;\n",
    "print(\"Bottom Boundary Condition\")\n",
    "nrtrue = 3*Lx;\n",
    "print(\"    %d rows, expected %d\" % (nr-nrstart, nrtrue ));\n",
    "netrue = 4*Lx;\n",
    "print(\"    %d elements, expected %d\" % (ne-nestart, netrue ));\n",
    "\n",
    "nrstart=nr;\n",
    "nestart=ne;\n",
    "# top z=0 boundary, dq/dz = d ux / dz = uz = 0\n",
    "iz=Lz-1;\n",
    "for ix in range(Lx):\n",
    "    k10q = kofijc[ix,iz-1,0];\n",
    "    k11q = kofijc[ix,iz,0];\n",
    "    k10x = kofijc[ix,iz-1,1];\n",
    "    k11x = kofijc[ix,iz,1];\n",
    "    k11z = kofijc[ix,iz,2];\n",
    "    Dr[ne,0] = nr;  # dq/dz = 0 (2 elements)\n",
    "    Dc[ne,0] = k10q;\n",
    "    Dv[ne,0] = -Dzr;\n",
    "    ne=ne+1;\n",
    "    Dr[ne,0] = nr;  \n",
    "    Dc[ne,0] = k11q;\n",
    "    Dv[ne,0] = Dzr;\n",
    "    ne=ne+1;\n",
    "    nr=nr+1;\n",
    "    Dr[ne,0] = nr; # d ux / dz = 0 (2 elements)\n",
    "    Dc[ne,0] = k10x;\n",
    "    Dv[ne,0] = -Dzr;\n",
    "    ne=ne+1;\n",
    "    Dr[ne,0] = nr;\n",
    "    Dc[ne,0] = k11x;\n",
    "    Dv[ne,0] = Dzr;\n",
    "    ne=ne+1;\n",
    "    nr=nr+1;\n",
    "    Dr[ne,0] = nr; # uz = 0 (1 element)\n",
    "    Dc[ne,0] = k11z;\n",
    "    Dv[ne,0] = 1.0;\n",
    "    ne=ne+1;\n",
    "    nr=nr+1;\n",
    "\n",
    "print(\"Top Boundary Condition\")\n",
    "nrtrue = 3*Lx;\n",
    "print(\"    %d rows, expected %d\" % (nr-nrstart, nrtrue ));\n",
    "netrue = 5*Lx;\n",
    "print(\"    %d elements, expected %d\" % (ne-nestart, netrue ));\n",
    "print(\"%d rows, %d expected\" % (nr,M));\n",
    "\n",
    "# solve in least squares sense wit bicg solver\n",
    "# Step 1: make sparse matrix\n",
    "L=M;\n",
    "Dsparse=sparse.coo_matrix((Dv[0:ne,0:1].ravel(),\n",
    "                             (Dr[0:ne,0:1].ravel(),Dc[0:ne,0:1].ravel())),shape=(L,M)).tocsr();\n",
    "del Dv; del Dr; del Dc;\n",
    "\n",
    "first=1;\n",
    "for myiter in range(1,NITER):\n",
    "\n",
    "    s = np.zeros((M,1));\n",
    "\n",
    "    nr = 0; # row counter\n",
    "    # interior equations, of which there are three\n",
    "    #          1:    dq/dx - d2 ux/ dx2 - d2 ux/ dz2 = 0\n",
    "    #          2:    dq/dz - d2 uz/ dx2 - d2 uz/ dz2 = Ra T\n",
    "    #                div u = 0\n",
    "    #          3:    d ux / dx + d uz / dz = 0\n",
    "    for ix in range(Lx):\n",
    "        for iz in range(1,Lz-1):\n",
    "            # 1: dq/dx - d2 ux/ dx2 - d2 ux/ dz2 = 0 (7 elements)\n",
    "            s[nr,0] = 0.0;\n",
    "            nr=nr+1;\n",
    "            # 2: dq/dz - d2 uz/ dx2 - d2 uz/ dz2 = Ra (T-Tlin)  (7 elements)\n",
    "            s[nr,0] = Ra * T[ix,iz];\n",
    "            nr=nr+1;\n",
    "            # 3: d ux / dx + d uz / dz = 0  (4 elements)\n",
    "            s[nr,0] = 0.0;\n",
    "            nr=nr+1;\n",
    "            \n",
    "    nrtrue = 3*(Lx-2)*(Lz-2);\n",
    "    netrue = 18*(Lx)*(Lz-2);\n",
    "    nrstart=nr;\n",
    "    nestart=ne;\n",
    "    # bottom z=0 boundary, q = ux = uz = 0\n",
    "    iz=0;\n",
    "    for ix in range(Lx):\n",
    "        s[nr,0] = 0.0;\n",
    "        nr=nr+1;\n",
    "        s[nr,0] = 0.0;\n",
    "        nr=nr+1;\n",
    "        s[nr,0] = 0.0;\n",
    "        nr=nr+1;\n",
    "\n",
    "    nrstart=nr;\n",
    "    nestart=ne;\n",
    "    # top z=0 boundary, dq/dz = ux = uz = 0\n",
    "    iz=Lz-1;\n",
    "    for ix in range(Lx):\n",
    "        s[nr,0] = 0.0;\n",
    "        nr=nr+1;\n",
    "        s[nr,0] = 0.0;\n",
    "        nr=nr+1;\n",
    "        s[nr,0] = 0.0;\n",
    "        nr=nr+1;\n",
    "        \n",
    "    # solve linear system\n",
    "    mtrue = gda_cvec(las.spsolve(Dsparse,s.ravel()));\n",
    "\n",
    "    Q = np.zeros((Lx,Lz));\n",
    "    UX = np.zeros((Lx,Lz));\n",
    "    UZ = np.zeros((Lx,Lz));\n",
    "    for ix in range(Lx):\n",
    "        for iz in range(Lz):\n",
    "            kq = kofijc[ix,iz,0];\n",
    "            kx = kofijc[ix,iz,1];\n",
    "            kz = kofijc[ix,iz,2];\n",
    "            Q[ix,iz]=mtrue[kq,0];\n",
    "            UX[ix,iz]=mtrue[kx,0];\n",
    "            UZ[ix,iz]=mtrue[kz,0];\n",
    "\n",
    "    e = s - Dsparse*mtrue;\n",
    "    sabsmax=np.max(np.max(s));\n",
    "    eabsmax=np.max(np.max(e));\n",
    "\n",
    "    # now advect\n",
    "    # dT/dt = Laplacian T - gradT dot u\n",
    "    # boundary condition T=0 at top and bottom, wrapping at sides\n",
    "    LAPT = np.zeros((Lx,Lz));\n",
    "    Tx = np.zeros((Lx,Lz));\n",
    "    Tz = np.zeros((Lx,Lz));\n",
    "    for ix in range(Lx):\n",
    "        for iz in range(1,Lz-1):\n",
    "            if( ix==0 ):\n",
    "                ix0 = Lx-1;\n",
    "                ix1 = ix;\n",
    "                ix2 = ix+1;\n",
    "            elif( ix==(Lx-1)):\n",
    "                ix0 = ix-1;\n",
    "                ix1 = ix;\n",
    "                ix2 = 0;\n",
    "            else:\n",
    "                ix0 = ix-1;\n",
    "                ix1 = ix;\n",
    "                ix2 = ix+1;\n",
    "            iz0 = iz-1;\n",
    "            iz1 = iz;\n",
    "            iz2 = iz+1;\n",
    "            d2Tdx2 = Dxr2*(T[ix0,iz1] - 2.0*T[ix1,iz1] + T[ix2,iz1] );\n",
    "            d2Tdz2 = Dzr2*(T[ix1,iz0] - 2.0*T[ix1,iz1] + T[ix1,iz2] );\n",
    "            LAPT[ix1,iz1] = d2Tdx2 + d2Tdz2;\n",
    "            Tx[ix1,iz1] = TDxr*( -T[ix0,iz1] + T[ix2,iz1] );\n",
    "            Tz[ix1,iz1] = TDzr*( -T[ix1,iz0] + T[ix1,iz2] );\n",
    "    Tx[0:Lx,0]=0.0;\n",
    "    Tx[0:Lx,Lz-1]=0.0;\n",
    "    Tz[0:Lx,0]=Tz[0:Lx,1];\n",
    "    Tz[0:Lx,Lz-1]=Tz[0:Lx,Lz-2] ; \n",
    "            \n",
    "    # Dt = 0.001;  # time increment\n",
    "    Dt = 0.00001;  # time increment\n",
    "\n",
    "    # temperature change\n",
    "    CONDUCT = 1.0;  # normally 1.0, but set to zero to supress conduction\n",
    "    ADVECT = 1.0;   # normally 1.0, but set to zero to supress advection\n",
    "    DT = Dt*(CONDUCT*LAPT - ADVECT*(np.multiply(Tx,UX) + np.multiply(Tz,UZ))); # interior\n",
    "    Tnew = T + DT;\n",
    "    \n",
    "    # smooth DTnew\n",
    "    Tnewsm = np.zeros((Lx,Lz));\n",
    "    for ix0 in range(Lx):\n",
    "        for iz0 in range(1,Lz-1):\n",
    "            for ix1 in [-1, 0, 1]:\n",
    "                for iz1 in [-1, 0, 1]:\n",
    "                    ix2 = (ix0+ix1)%Lx\n",
    "                    iz2 = (iz0+iz1);\n",
    "                    Tnewsm[ix0,iz0] = Tnewsm[ix0,iz0] + w[ix1+1,iz1+1] * Tnew[ix2,iz2];\n",
    "    Tnew=Tnewsm;\n",
    "    Tnew[0:Lx,0] =    Tbot;  # bottom BC\n",
    "    Tnew[0:Lx,Lz-1] = 0.0;   # top BC\n",
    "    T=Tnew;\n",
    "    TIMG[0:Lx,0:Lz,myiter]=T[0:Lx,0:Lz];\n",
    "    if(first or (not(myiter%SNAPSHOT))):\n",
    "        print('iter: ', myiter,\"max(abs(s-Dm)) / max(abs(s)): %e\" % (eabsmax/sabsmax));\n",
    "        gda_draw('title T',np.flipud(T.T),\n",
    "                 'title Q',np.flipud(Q.T),\n",
    "                 'title Ux',np.flipud(UX.T),\n",
    "                 'title Uz',np.flipud(UZ.T) );\n",
    "    first=0;\n",
    "     # save movie\n",
    "    TUX[0:Lx,0:Lz,myiter] = UX;\n",
    "    TUZ[0:Lx,0:Lz,myiter] = UZ;\n",
    "\n",
    "print('done with ',myiter+1,' iterations');\n",
    "\n",
    "fig = plt.figure();\n",
    "ax = fig.subplots();\n",
    "\n",
    "ax.set_xlim(( 0.0, xmax));\n",
    "ax.set_ylim(( 0.0, zmax));\n",
    "\n",
    "IMG = TIMG[0:Lx,0:Lz,0];\n",
    "dmin = np.min(TIMG);\n",
    "dmax = np.max(TIMG);\n",
    "mycmap=cm.jet;\n",
    "left=0; right=xmax;\n",
    "top=zmax; bottom=0;\n",
    "plt.imshow(np.flipud(IMG.T), cmap=mycmap, vmin=dmin, vmax=dmax, extent=(left,right,bottom,top) );\n",
    "im = ax.imshow([[]], cmap=mycmap, vmin=dmin, vmax=dmax, extent=(left,right,bottom,top) );\n",
    "plt.xlabel('x');\n",
    "plt.ylabel('y');\n",
    "plt.colorbar();\n",
    "\n",
    "def init():\n",
    "    im.set_data([[]]);\n",
    "    return (im,)\n",
    "\n",
    "def animate(i):\n",
    "    IMG = TIMG[0:Lx,0:Lz,i];\n",
    "    im.set_data(np.flipud(IMG.T) );\n",
    "    return (im,)\n",
    "\n",
    "anim = animation.FuncAnimation(fig, animate, init_func=init,\n",
    "                               frames=NITER, interval=50, \n",
    "                               blit=True)\n",
    "\n",
    "HTML(anim.to_html5_video())\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "writing File c:/bill/CES23/convectionanimation.mp4\n",
      "done\n"
     ]
    },
    {
     "data": {
      "image/png": "[encoded data, 56616 characters]",
      "text/plain": [
       "<Figure size 640x480 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# particle movie\n",
    "\n",
    "DEC = 5;  # decimation factor\n",
    "xpp = x[0:Lx:DEC,0:1]; # particle positions in x-direction\n",
    "zpp = z[0:Lz:DEC,0:1]; # particle positions in z-direction\n",
    "Nxp, i = np.shape(xpp); # number of particles in x direction\n",
    "Nzp, i = np.shape(zpp); # number of particles in z direction\n",
    "Np = Nxp*Nzp; # number of paricles\n",
    "xp = np.zeros((Np,1)); # x position of particles\n",
    "zp = np.zeros((Np,1)); # z position of particles\n",
    "k=0;\n",
    "for i in range(Nxp):\n",
    "    for j in range(Nzp):\n",
    "        xp[k,0] = xpp[i,0];\n",
    "        zp[k,0] = zpp[j,0];\n",
    "        k=k+1;\n",
    "\n",
    "fig = plt.figure();\n",
    "ax = fig.subplots();\n",
    "\n",
    "ax.set_xlim((0.0, xmax));\n",
    "ax.set_ylim((0.0, zmax));\n",
    "\n",
    "\n",
    "IMG = TIMG[0:Lx,0:Lz,0];\n",
    "dmin = np.min(TIMG);\n",
    "dmax = np.max(TIMG);\n",
    "mycmap=cm.jet;\n",
    "left=0; right=xmax;\n",
    "top=zmax; bottom=0;\n",
    "plt.imshow(np.flipud(IMG.T), cmap=mycmap, vmin=dmin, vmax=dmax, extent=(left,right,bottom,top) );\n",
    "im = ax.imshow([[]], cmap=mycmap, vmin=dmin, vmax=dmax, extent=(left,right,bottom,top) );\n",
    "plt.xlabel('x');\n",
    "plt.ylabel('y');\n",
    "plt.colorbar();\n",
    "\n",
    "line1, = ax.plot(zp, zp, 'w.', lw=1);\n",
    "\n",
    "def init():\n",
    "    line1.set_data([], []);\n",
    "    im.set_data([[]]);\n",
    "    return (im,line1)\n",
    "\n",
    "def animate(i):\n",
    "    IMG = TIMG[0:Lx,0:Lz,i];\n",
    "    im.set_data(np.flipud(IMG.T) );\n",
    "    for j in range(Np):\n",
    "        ix = int(xp[j,0]/Dx); # x index of particle j\n",
    "        if( ix<0 ):\n",
    "            ix=0;\n",
    "        elif( ix>(Lx-1) ):\n",
    "            ix = Lx-1;\n",
    "        iz = int(zp[j,0]/Dz); # z index of particle j\n",
    "        if( iz<0 ):\n",
    "            ix=0;\n",
    "        elif( iz>(Lz-1) ):\n",
    "            iz = Lz-1;\n",
    "        uxp = TUX[ix,iz,i]; # velocity of particle\n",
    "        uzp = TUZ[ix,iz,i];\n",
    "        xp[j,0] = xp[j,0] + uxp*Dt; # new positioon of particle\n",
    "        zp[j,0] = zp[j,0] + uzp*Dt;\n",
    "        if( xp[j,0]<0.0 ): # wrapping at ends\n",
    "             xp[j,0]= xp[j,0] + xmax;\n",
    "        elif( xp[j,0]>xmax ):\n",
    "             xp[j,0]= xp[j,0] - xmax;\n",
    "        if( zp[j,0]<0.0 ):\n",
    "            zp[j,0]=0.0;\n",
    "        elif( zp[j,0]>zmax ):\n",
    "            zp[j,0] = zmax;\n",
    "    line1.set_data(xp, zp);\n",
    "    return (im,line1)\n",
    "\n",
    "anim = animation.FuncAnimation(fig, animate, init_func=init,\n",
    "                               frames=NITER, interval=20, \n",
    "                               blit=True)\n",
    "\n",
    "f = \"c:/bill/CES23/Convection/convectionanimation.mp4\" ;\n",
    "print(\"writing File %s\" % (f) );\n",
    "writervideo = animation.FFMpegWriter(fps=60) \n",
    "anim.save(f, writer=writervideo)\n",
    "print(\"done\");\n",
    "\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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