{
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   "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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   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
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   "outputs": [
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     "metadata": {},
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    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "elements:  32319  expected  32319\n",
      "model parameters:\n",
      "top/bottom bounrary:          81 81  expected  81 81\n",
      "volume:                       6399  expected  6399\n",
      "total  6561  expected  6561\n"
     ]
    },
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    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "done with  1499  iterations\n"
     ]
    },
    {
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     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
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       "<Figure size 640x480 with 2 Axes>"
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   "source": [
    "# flow in a 2D Cartesian conduit with one moving boundary and wrapping at ends\n",
    "\n",
    "# here is a synopysis of the theory\n",
    "\n",
    "# boundary conditions along conduit\n",
    "#     pressure: dq/dz(x,z=0)=0  and  dq/dz(x,z=zmax)=0\n",
    "#     x-velocity: dUX/dx(x,z=0)=0 and dUZ/dx(x,z=zmax)=u0\n",
    "#     z-velocity: UZ(x,z=0)=0 and UZ(x,z=zmax)=0\n",
    "\n",
    "# In a incompressible  fluid, the stress is a combination of shear stress and pressure\n",
    "# stress = shear stress - presure = s_ij = tau_ij - p delta_ij\n",
    "# div stress = s_ij,j = tau_ij,j - p,i\n",
    "# impompressible viscous fluid:  tau_ij = v' ( u_i,j + u_j,i )  with viscosity v' and  u_j,j=0\n",
    "# div shear stress = tau_ij,j = v' ( u_i,jj + u_j,ij )\n",
    "# but note u_j,ij = (u_j,j)_,i = 0 by continuity condition, u_i,i=0, so\n",
    "# tau_ij,j = v' u_i,jj = v' Laplacian u\n",
    "\n",
    "# Conservation of momentum leads to the Navier-Stokes equation\n",
    "# change in momentum = momentum flux + contact force + body force\n",
    "# (d/dt) rho u_i = -(rho u_i u_j ),j - p_,i + tau_ij,j + f_i\n",
    "# (d/dt) rho u_i = -(rho u_i u_j ),j - p_,i + v' u_i,jj + f_i           but we will assume f_i=0\n",
    "# (d/dt) u_i = -(u_i u_j ),j - (p/rho)_,i + v u_i,jj + (f/rho)_i        with v = v'/rho = kinematic viscosity\n",
    "\n",
    "# note, one term can be somplified\n",
    "# (u_i u_j ),j = u_i,j u_j + u_i (u_j,j) = u_i,j u_j as u_j,j=0\n",
    "\n",
    "# so\n",
    "# (d/dt) u_i = -u_i,j u_j - (p/rho)_,i + v u_i,jj  (matches Menke & Abbott, Eqn. 11.1.4)\n",
    "\n",
    "# we will solve using finite differences using an explicit scheme\n",
    "# here velocoty u at time n+1 is denoted u(n+1).  The NS equation is\n",
    "# u(n+1)_i - un_i = Dt ( - un_i,j un_j - (pn/rho)_,i + v un_i,jj )\n",
    "\n",
    "# the big issue is how to determine the pressure.  We use a standard trick in\n",
    "# which the pressure is chosen to make u(n+1) (velocity at the next time increment)\n",
    "# satisfy the continuity condition. We take divergence of the NS equation\n",
    "# u(n+1)_i,i  - un_i,i = Dt ( -un_i,ij un_j -un_i,j un_j,i - (pn/rho)_,ii + v un_i,jji )\n",
    "# assume un has zero divergence\n",
    "#  u(n+1)_i,i -    0   = Dt (     0       -un_i,j un_j,i   - (pn/rho)_,ii +    0\n",
    "# to enforce u(n+1) has zero divergence\n",
    "# qn_ii = (pn/rho)_,ii = -  un_i,j un_j,i )\n",
    "# or Poisson's equation for scaled pressure q\n",
    "# qn_,ii = Sn  with   Sn = - un_i,j un_j,i\n",
    "# Sn = -( (un_1,1)**2 + (un_2,2)**2 + 2 (un_1,2*un_2,1) )\n",
    "\n",
    "# As only grad q appears in the NS equation, the pressure is determined\n",
    "# only up to an additive constant.  I ignore this issue and set up and\n",
    "# solve the pressure equation without any constraints. The equation is\n",
    "# rank deficient, but the solver seems to find a solution anyway.  I then\n",
    "# subtract out the mean and add back in a known constant.\n",
    "\n",
    "# to handle the possibility that div un is not exactly xero\n",
    "# one could add the term Spn = -un_i,ij un_j = -divn_,j un_j to Sn\n",
    "# Spn = -( divn_,1 un_1 + divm_,2 un_2 ) \n",
    "\n",
    "# furthermore, one could add the term Sppn = v un_i,jji = v (un_i,i)_,jj=  v laplacoan div un\n",
    "\n",
    "# furthermore, one could add the term Spppn = un_i,i/Dt = divn/Dt\n",
    "\n",
    "# The code plots the rms div u as a function of time, and also allows one to make a movie of\n",
    "# div u.  It's not exactly zero, but it it small and seems to be well behaved.\n",
    "\n",
    "# so one solves Poisson's equation to obtain the scaled-pressure field qn that will enforce div u(n+1) = 0\n",
    "# and then one advanece time via\n",
    "# u(n+1)_i = un_i + Dt ( -un_i,j un_j - qn_,i + v un_i,jj )\n",
    "# u(n+1)_i = un_i + Dt ( Hn_i - qn_i )  with \n",
    "# Hn_i = -u_i,j u_j + v un_i,jj\n",
    "# Hn_1 = -u_1,1 u_1  -u_1,2 u_2 + v un_1,11 + v un_1,22\n",
    "# Hn_2 = -u_2,1 u_1  -u_2,2 u_2 + v un_2,11 + v un_2,22\n",
    "\n",
    "# at each time interval, the bounary conditions on velocity must be enforced\n",
    "# by explicitly resetting some value of the velocity grid\n",
    "\n",
    "# in the case of stationary horizontal flow with no perturbations, uz=0 for all x,z\n",
    "# implting (d/dz uz=0) so continuity, d/dx ux + d/dz uz = 0, inplies d/dx ux = 0,\n",
    "# which is to say ux varies only wih z.  The NS equations come\n",
    "# dp/dx = v' (d2/dz2) ux and dp/dz=0, so p is only a function of x\n",
    "# so the lhs of dp/dx = v' (d2/dz2) ux is only a function of x and the rhs of z, and\n",
    "# dp/dx = v' (d2/dz2) ux = C = constant.  A valid solution is C=0, in which case\n",
    "# ux = ux0 z / zmax and p=constant=p0 where p0 is arbitrary\n",
    "\n",
    "# overall parzameters\n",
    "UX0 = 1.0;   # velocity of top boundary\n",
    "p0 = 1.0;   # background pressure\n",
    "vp = 0.001;  # viscosity\n",
    "rho = 1.0;  # density\n",
    "v = vp/rho; # kinematic viscosity\n",
    "q0 = p0/rho; # scaled pressure\n",
    "f = 2.0; # fractional perturbation in velocity\n",
    "direction = 1.0; # direction of spin of perturbation\n",
    "Dt = 1e-3; # tim increment\n",
    "\n",
    "NITER=1500;\n",
    "ITYPE = 1;    # Movie variable,  1: |v|, 2: div\n",
    "\n",
    "# x-axis\n",
    "Lx = 81;\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",
    "Dxr2 = 1.0/(Dx**2);\n",
    "TDxr = 1.0/(2.0*Dx);\n",
    "\n",
    "# z-axis\n",
    "Lz = 81;\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",
    "Dzr2 = 1.0/(Dz**2);\n",
    "TDzr = 1.0/(2.0*Dz);\n",
    "\n",
    "# the movie\n",
    "TIMG=np.zeros((Lx,Lz,NITER));\n",
    "TUX=np.zeros((Lx,Lz,NITER));\n",
    "TUZ=np.zeros((Lx,Lz,NITER));\n",
    "\n",
    "# keek track of divergence\n",
    "RMSDIV = np.zeros((NITER,1));\n",
    " \n",
    "# initial velocity field: horizontal flow plus a circular perturbation with tangential velocity\n",
    "x0 = (xmax-Dx/2.0)/2.0;\n",
    "z0 = (xmax-Dx/2.0)/2.0;\n",
    "sigma = 0.07;\n",
    "sigma2 = sigma**2;\n",
    "Q0 = q0*np.ones((Lx,Lz));\n",
    "Q0X = 0.0;\n",
    "UX = (UX0/zmax)*np.matmul( np.ones((Lx,1)), z.T );\n",
    "UX00=np.copy(UX);\n",
    "UZ = np.zeros((Lx,Lz));\n",
    "for ix in range(Lx):\n",
    "    DX = x[ix,0]-x0;\n",
    "    for iz in range(Lz):\n",
    "        DZ = z[iz,0]-z0;\n",
    "        R2 = DX**2 + DZ**2;\n",
    "        R = sqrt(R2);\n",
    "        # Actual perturbation is\n",
    "        # A = (f*UX0*R**2/sigma2)*exp(-0.5*R2/sigma2); and t = (DZ,-DX)/R\n",
    "        # but camcel factor of R to avoid division by zero error\n",
    "        A = (f*UX0*R/sigma2)*exp(-0.5*R2/sigma2);\n",
    "        tx =  direction*DZ; # tangential udirection\n",
    "        tz = -direction*DX;\n",
    "        UX[ix,iz] = UX[ix,iz] + A*tx;\n",
    "        UZ[ix,iz] = UZ[ix,iz] + A*tz;\n",
    "        \n",
    "gda_draw('title Q0', Q0, 'title UX', UX, 'title UZ', UZ);\n",
    "TUX[0:Lx,0:Lz,0] = UX;\n",
    "TUZ[0:Lx,0:Lz,0] = UZ;\n",
    "if( ITYPE==1 ):\n",
    "    TIMG[0:Lx,0:Lz,0] = np.sqrt(np.power(UX,2)+np.power(UZ,2));\n",
    "elif( ITYPE==2 ):\n",
    "    TIMG[0:Lx,0:Lz,0] = np.zeros((Lx,Lz));\n",
    "\n",
    "# setup for Poisson Equation\n",
    "\n",
    "# lookup tables translate between (i,j) image an linear k ordering\n",
    "M = Lx*Lz; # number of unknowns in pressure equation\n",
    "iofk = np.zeros((M,1),dtype=int); # x position\n",
    "jofk = np.zeros((M,1),dtype=int); # y position\n",
    "kofij = np.zeros((Lx,Lz),dtype=int);\n",
    "k=0;\n",
    "for ix in range(Lx):\n",
    "    for iz in range(Lz):\n",
    "        iofk[k,0]=ix;\n",
    "        jofk[k,0]=iz;\n",
    "        kofij[ix,iz]=k;\n",
    "        k=k+1;\n",
    "        \n",
    "Pmax = 100000;\n",
    "Dr = np.zeros((Pmax,1),dtype=int);\n",
    "Dc = np.zeros((Pmax,1),dtype=int);\n",
    "Dv = np.zeros((Pmax,1));\n",
    "s = np.zeros((M,1));\n",
    "\n",
    "# set up matrix\n",
    "nr=0;\n",
    "ne=0;\n",
    "meb=0; met=0; mi=0;\n",
    "for ix in range(Lx):\n",
    "    for iz in range(Lz):\n",
    "        if( iz==0 ):\n",
    "            k=kofij[ix,iz]; # bottom boundary, excluding first point\n",
    "            kp1=kofij[ix,iz+1];\n",
    "            meb=meb+1;\n",
    "            Dr[ne,0] = nr;\n",
    "            Dc[ne,0] = k;\n",
    "            Dv[ne,0] = -1.0;\n",
    "            ne=ne+1;\n",
    "            Dr[ne,0] = nr;\n",
    "            Dc[ne,0] = kp1;\n",
    "            Dv[ne,0] = 1.0;\n",
    "            ne=ne+1\n",
    "            nr=nr+1;\n",
    "        elif( (iz==(Lz-1)) ):\n",
    "            k=kofij[ix,iz]; # top boundary\n",
    "            km1=kofij[ix,iz-1]; \n",
    "            met=met+1;\n",
    "            Dr[ne,0] = nr;\n",
    "            Dc[ne,0] = k;\n",
    "            Dv[ne,0] = 1.0;\n",
    "            ne=ne+1;\n",
    "            Dr[ne,0] = nr;\n",
    "            Dc[ne,0] = km1;\n",
    "            Dv[ne,0] = -1.0;\n",
    "            ne=ne+1;\n",
    "            nr=nr+1;\n",
    "        else: # interior\n",
    "            # wrapping in x\n",
    "            if( ix==0 ):\n",
    "                ixm = Lx-1;\n",
    "                ixp = ix+1;\n",
    "            elif( ix==(Lx-1) ):\n",
    "                ixm = ix-1;\n",
    "                ixp = 0;\n",
    "            else:\n",
    "                ixm = ix-1;\n",
    "                ixp = ix+1;\n",
    "            k00=kofij[ixm,iz-1]; # interior\n",
    "            k01=kofij[ixm,iz];\n",
    "            k02=kofij[ixm,iz+1];\n",
    "            k10=kofij[ix,iz-1];\n",
    "            k11=kofij[ix,iz];\n",
    "            k12=kofij[ix,iz+1];\n",
    "            k20=kofij[ixp,iz-1];\n",
    "            k21=kofij[ixp,iz];\n",
    "            k22=kofij[ixp,iz+1];\n",
    "            mi=mi+1;\n",
    "            # Laplacian d2 T / dx2 + d2 T / dy2 = 0\n",
    "            # d2 T / dx2\n",
    "            Dr[ne,0] = nr;\n",
    "            Dc[ne,0] = k01;\n",
    "            Dv[ne,0] = Dxr2;\n",
    "            ne=ne+1;\n",
    "            Dr[ne,0] = nr;\n",
    "            Dc[ne,0] = k11;\n",
    "            Dv[ne,0] = -2.0*Dxr2 -2.0*Dzr2;\n",
    "            ne=ne+1;\n",
    "            Dr[ne,0] = nr;\n",
    "            Dc[ne,0] = k21;\n",
    "            Dv[ne,0] = Dxr2;\n",
    "            ne=ne+1;\n",
    "            # d2 T / dy2\n",
    "            Dr[ne,0] = nr;\n",
    "            Dc[ne,0] = k10;\n",
    "            Dv[ne,0] = Dzr2;\n",
    "            ne=ne+1;\n",
    "            # Dr[ne,0] = nr; Absorbed into earlie element\n",
    "            # Dc[ne,0] = k11;\n",
    "            # Dv[ne,0] = -2.0*Dzr2;\n",
    "            # ne[ne,0]=ne+1;\n",
    "            Dr[ne,0] = nr;\n",
    "            Dc[ne,0] = k12;\n",
    "            Dv[ne,0] = Dzr2;\n",
    "            ne=ne+1;\n",
    "            nr=nr+1;\n",
    "\n",
    "print('elements: ',ne,' expected ',4*Lx+5*Lx*(Lz-2) );\n",
    "print('model parameters:')\n",
    "print('top/bottom bounrary:         ', met, meb, ' expected ', Lx, Lx );\n",
    "print('volume:                      ', mi, ' expected ', Lx*(Lz-2));\n",
    "print('total ', meb+met+mi, ' expected ', M);\n",
    "\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=(M,M)).tocsr();\n",
    "del Dv; del Dr; del Dc;\n",
    "\n",
    "first=1;\n",
    "for myiter in range(1,NITER):\n",
    "    # derivatives of the velocity field\n",
    "    UXX = np.zeros((Lx,Lz));\n",
    "    UZZ = np.zeros((Lx,Lz));\n",
    "    UXZ = np.zeros((Lx,Lz));\n",
    "    UZX = np.zeros((Lx,Lz));\n",
    "    LAPUX = np.zeros((Lx,Lz));\n",
    "    LAPUZ = np.zeros((Lx,Lz));\n",
    "    HX = np.zeros((Lx,Lz));\n",
    "    HZ = np.zeros((Lx,Lz));\n",
    "    S = np.zeros((Lx,Lz));\n",
    "    for ix in range(Lx):\n",
    "        # wrapping in x\n",
    "        if( ix==0 ):\n",
    "            ixm = Lx-1;\n",
    "            ixp = ix+1;\n",
    "        elif( ix==(Lx-1) ):\n",
    "            ixm = ix-1;\n",
    "            ixp = 0;\n",
    "        else:\n",
    "            ixm = ix-1;\n",
    "            ixp = ix+1;\n",
    "        for iz in range(1,Lz-1):\n",
    "            UXX[ix,iz] = TDxr * (UX[ixp,iz] - UX[ixm,iz]);\n",
    "            UZZ[ix,iz] = TDzr * (UZ[ix,iz+1] - UZ[ix,iz-1]);\n",
    "            UXZ[ix,iz] = TDzr * (UX[ix,iz+1] - UX[ix,iz-1]);\n",
    "            UZX[ix,iz] = TDxr * (UZ[ixp,iz] - UZ[ixm,iz]);\n",
    "            LAPUX[ix,iz] = ( Dxr2 * (UX[ixp,iz] - 2.0*UX[ix,iz] + UX[ixm,iz] )\n",
    "                         +   Dzr2 * (UX[ix,iz+1] - 2.0*UX[ix,iz] + UX[ix,iz-1] ) );\n",
    "            LAPUZ[ix,iz] = ( Dxr2 * (UZ[ixp,iz] - 2.0*UZ[ix,iz] + UZ[ixm,iz] )\n",
    "                         +   Dzr2 * (UZ[ix,iz+1] - 2.0*UZ[ix,iz] + UZ[ix,iz-1] ) );\n",
    "    # Hn_1 = -u_1,1 u_1  -u_1,2 u_2 + v un_1,11 + v un_1,22\n",
    "    # Hn_2 = -u_2,1 u_1  -u_2,2 u_2 + v un_2,11 + v un_2,22\n",
    "    HX = -np.multiply(UXX,UX) - np.multiply(UXZ,UZ) + v * LAPUX;\n",
    "    HZ = -np.multiply(UZX,UX) - np.multiply(UZZ,UZ) + v * LAPUZ;\n",
    "    DIVU = UXX+UZZ;\n",
    "    RMSDIV[myiter,0] = sqrt(np.sum(np.power(DIVU,2))/(Lx*Lz));\n",
    "    # source Sn = -( (un_1,1)**2 + (un_2,2)**2 + 2 (un_1,2*un_2,1) )\n",
    "    S = -( np.power(UXX,2) + np.power(UZZ,2) + 2.0*np.multiply(UXZ,UZX) );\n",
    "    if( 1 ):\n",
    "        #Spn = -( divn_,1 un_1 + divm_,2 un_2 ) \n",
    "        DIVUX = np.zeros((Lx,Lz));\n",
    "        DIVUZ = np.zeros((Lx,Lz));\n",
    "        for ix in range(Lx):\n",
    "            # wrapping in x\n",
    "            if( ix==0 ):\n",
    "                ixm = Lx-1;\n",
    "                ixp = ix+1;\n",
    "            elif( ix==(Lx-1) ):\n",
    "                ixm = ix-1;\n",
    "                ixp = 0;\n",
    "            else:\n",
    "                ixm = ix-1;\n",
    "                ixp = ix+1;\n",
    "            for iz in range(1,Lz-1):\n",
    "                DIVUX[ix,iz] = TDxr * (DIVU[ixp,iz] - DIVU[ixm,iz]);\n",
    "                DIVUZ[ix,iz] = TDzr * (DIVU[ix,iz+1] - DIVU[ix,iz-1]);\n",
    "        Sp = -( np.multiply(DIVUX,UX) + np.multiply(DIVUZ,UZ) );\n",
    "        S = S+Sp;\n",
    "    if( 1 ):\n",
    "        # Sppn =  v laplacoan div un\n",
    "        LAPDIVU = np.zeros((Lx,Lz));\n",
    "        for ix in range(Lx):\n",
    "            # wrapping in x\n",
    "            if( ix==0 ):\n",
    "                ixm = Lx-1;\n",
    "                ixp = ix+1;\n",
    "            elif( ix==(Lx-1) ):\n",
    "                ixm = ix-1;\n",
    "                ixp = 0;\n",
    "            else:\n",
    "                ixm = ix-1;\n",
    "                ixp = ix+1;\n",
    "            for iz in range(1,Lz-1):\n",
    "                LAPDIVU[ix,iz] = ( Dxr2 * (DIVU[ixp,iz] - 2.0*DIVU[ix,iz] + DIVU[ixm,iz] )\n",
    "                               +   Dzr2 * (DIVU[ix,iz+1] - 2.0*DIVU[ix,iz] + DIVU[ix,iz-1] ) );\n",
    "        Spp = v * LAPDIVU;\n",
    "        S = S+Spp;\n",
    "    if( 1 ):\n",
    "        # Spppn = divn / Dt\n",
    "        Sppp = DIVU / Dt;\n",
    "        S = S+Sppp;\n",
    "    if(first):\n",
    "        gda_draw('title HX', HX, 'title HZ', HZ, 'title S', S, 'title divU', DIVU);\n",
    "\n",
    "    # set up right-hand side\n",
    "    nr=0;\n",
    "    ne=0;\n",
    "    mel=0; meb=0; met=0; mi=0;\n",
    "    for ix in range(Lx):\n",
    "        for iz in range(Lz):\n",
    "            if( iz==0 ):\n",
    "                k=kofij[ix,iz]; # bottom boundary excl first pt\n",
    "                s[k,0]=0.0;\n",
    "                nr=nr+1;\n",
    "            elif( (iz==(Lz-1)) ):\n",
    "                k=kofij[ix,iz]; # top exterior boundary\n",
    "                s[k,0]=0.0;\n",
    "                nr=nr+1;\n",
    "            else:\n",
    "                k11=kofij[ix,iz];\n",
    "                s[k11,0]=S[ix,iz];\n",
    "                nr=nr+1;\n",
    "\n",
    "    # solve for q.\n",
    "    # I'm surprized this works, becase Dsparse is ramk deficcient;\n",
    "    # as it is missing a constraint on the mean value\n",
    "    mest = gda_cvec(las.spsolve(Dsparse,s.ravel()));\n",
    "    \n",
    "    # the pressure is determined only up to a constant, so demean it and add back a known constant\n",
    "    mest = mest-np.mean(mest) + q0;\n",
    "\n",
    "    # fold into image\n",
    "    Q = np.zeros((Lx,Lz));\n",
    "    for ix in range(Lx):\n",
    "        for iz in range(Lz):\n",
    "            k=kofij[ix,iz];\n",
    "            Q[ix,iz] = mest[k,0];\n",
    "    if(first):\n",
    "        gda_draw('title Q', Q,'title Q-Q0', Q-Q0);\n",
    "\n",
    "    # pressure gradient\n",
    "    QX = np.zeros((Lx,Lz));\n",
    "    QZ = np.zeros((Lx,Lz));\n",
    "    for ix in range(Lx):\n",
    "        # wrapping in x\n",
    "        if( ix==0 ):\n",
    "            ixm = Lx-1;\n",
    "            ixp = ix+1;\n",
    "        elif( ix==(Lx-1) ):\n",
    "            ixm = ix-1;\n",
    "            ixp = 0;\n",
    "        else:\n",
    "            ixm = ix-1;\n",
    "            ixp = ix+1;\n",
    "        for iz in range(1,Lz-1):\n",
    "            QX[ix,iz] = TDxr * (Q[ixp,iz] - Q[ixm,iz]);\n",
    "            QZ[ix,iz] = TDzr * (Q[ix,iz+1] - Q[ix,iz-1]);\n",
    "    if(first):\n",
    "        gda_draw('title QX',QX,'title QZ',QZ);\n",
    "\n",
    "    DUX = Dt * ( HX - QX );\n",
    "    DUZ = Dt * ( HZ - QZ );\n",
    "\n",
    "    if(first):\n",
    "        gda_draw('title DUZ',DUX,'title DUZ',DUZ);\n",
    "\n",
    "    # time update\n",
    "    UX = UX + DUX;\n",
    "    UZ = UZ + DUZ;\n",
    "    \n",
    "    # enforce boundary conditions\n",
    "    UX[0:Lx,0] = 0.0;\n",
    "    UZ[0:Lx,0] = 0.0;\n",
    "    UX[0:Lx,Lz-1] = UX0;\n",
    "    UZ[0:Lx,Lz-1] = 0.0;\n",
    "\n",
    "    # save movie\n",
    "    TUX[0:Lx,0:Lz,myiter] = UX;\n",
    "    TUZ[0:Lx,0:Lz,myiter] = UZ;\n",
    "    if( ITYPE == 1 ):\n",
    "        TIMG[0:Lx,0:Lz,myiter] = np.sqrt(np.power(UX,2)+np.power(UZ,2));\n",
    "    elif( ITYPE == 2 ):\n",
    "        TIMG[0:Lx,0:Lz,myiter] = DIVU;\n",
    "    first=0;\n",
    "\n",
    "print('done with ',myiter,' iterations');\n",
    "fig = plt.figure();\n",
    "ax = plt.subplot(1,1,1);\n",
    "plt.axis( [0, NITER, 0, np.max(RMSDIV)] );\n",
    "plt.plot( gda_cvec(np.linspace(0,NITER-1,NITER)), RMSDIV, 'k-' );\n",
    "plt.xlabel('time');\n",
    "plt.ylabel('rms div velocity');\n",
    "plt.show();\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=20, \n",
    "                               blit=True)\n",
    "\n",
    "HTML(anim.to_html5_video())\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
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       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "[encoded data, 22832 characters]",
      "text/plain": [
       "<Figure size 640x480 with 1 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",
    "line1, = ax.plot([], [], 'k.', lw=1);\n",
    "\n",
    "def init():\n",
    "    line1.set_data(xp, zp);\n",
    "    return (line1,)\n",
    "\n",
    "def animate(i):\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 (line1,)\n",
    "\n",
    "anim = animation.FuncAnimation(fig, animate, init_func=init,\n",
    "                               frames=NITER, interval=20, \n",
    "                               blit=True)\n",
    "\n",
    "HTML(anim.to_html5_video())\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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   "name": "python3"
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    "name": "ipython",
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   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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