Commit ba198e97 by Mathieu RASSON

### Complete solver

parent ffd56e4b
 { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Crank-Nicholson scheme" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Cylindrical Diffraction Term" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "\"\"\"\n", "### Adapted crank Nicholson class.\n", "\n", "* Prepare half-step method.\n", "* Prepare non linear term for one of the half-step.\n", "* Use \$V\$ advection to add first radial derivative.\n", "\n", "\$V\$ is a function of \$r\$\n", "\$\$\n", "V(r) = \\dfrac{i}{2 k_0} \\dfrac{1}{r}\n", "\$\$\n", "and \$V(0) = 0\$.\n", "**Warning**: do not use \$V(0)=0\$ for any calculation.\n", "\n", "**Warning**: Test scheme matrices separately.\n", "\"\"\"\n", "\n", "import numpy as np\n", "import scipy.sparse\n", "import scipy.linalg as la\n", "\n", "class CrankNicolson:\n", " \"\"\"A class that solves dE/dz = D*d2E/dr2 + V(r)*dE/dr\"\"\"\n", " \n", " # Cylindrical grid\n", " def set_grid(self, r_max, n_r, z_min, z_max, n_z):\n", "\n", " self.r_max, self.n_r = r_max, n_r\n", " self.z_min, self.z_max, self.n_z = z_min, z_max, n_z\n", " self.r_pts, self.delta_r = np.linspace(0, r_max, n_r, retstep=True, endpoint=False)\n", " self.z_pts, self.delta_z = np.linspace(z_min, z_max, n_z, retstep=True, endpoint=False)\n", " \n", " # Parameters of the scheme\n", " def set_parameters(self, D, V):\n", " \n", " # V has to be vectorised\n", " self.D, self.V = D, V\n", " \n", " \n", " \n", " # One solving step for r dependency\n", " def solve(self, E_init):\n", " \n", " # Coefficient of matrices\n", " sig = self.D * self.delta_z / 2. / self.delta_r**2\n", " nu = lambda x: - self.delta_z / 4. / self.delta_r * self.V(x) # minus sign for V convention\n", " \n", " # Empty solution matrix\n", " self.E_matrix = np.zeros([self.n_z, self.n_r], dtype=complex)\n", " \n", " # Sparse solver\n", " A = self._fillA_sp(sig, nu, self.n_r)\n", " B = self._fillB_sp(sig, nu, self.n_r)\n", " \n", " # Set boundary conditions \n", " # Dirichlet at infinity\n", " A[1,-1] = 1.0\n", " A[2,-2] = 0.0\n", " B[-1,-1] = 0.0\n", " B[-1,-2] = 0.0\n", " # Neumann at r=0\n", " A[0,1] = -2*sig\n", " B[0,1] = 2*sig\n", " \n", " # Propagate\n", " E = E_init.copy()\n", " for n in range(self.n_z):\n", " self.E_matrix[n,:] = E\n", " \n", " E = la.solve_banded((1,1),A, B.dot(E), check_finite=False)\n", " \n", " \n", " \n", " # Deep copy of results for export\n", " # np.savetxt ?\n", " def get_E(self):\n", " \n", " return self.E_matrix.copy()\n", " \n", " # Diagonal packing for banded\n", " def _fillA_sp(self, sig, nu, n):\n", " \"\"\"Returns a tridiagonal matrix in compact form ab[1+i-j,j]=a[i,j]\"\"\"\n", " \n", " A = np.zeros([3,n], dtype=complex) # A has three diagonals and size n\n", " # Superdiagonal\n", " A[0,1:] = -(sig - nu(self.r_pts[:-1]))\n", " # Diagonal\n", " A[1] = 1+2*sig\n", " # Subdiagonal\n", " A[2,:-1] = -(sig + nu(self.r_pts[1:]))\n", " \n", " return A\n", " \n", " # Sparse tridiagonal storage\n", " def _fillB_sp(self, sig, nu, n):\n", " \"\"\"Returns a tridiagonal sparse matrix in csr-form\"\"\"\n", " \n", " _o = np.ones(n, dtype=complex)\n", " supdiag = (sig - nu(self.r_pts[:-1]))\n", " diag = (1-2*sig)*_o\n", " subdiag = (sig + nu(self.r_pts[1:]))\n", " \n", " return scipy.sparse.diags([supdiag, diag, subdiag], [1,0,-1], (n,n), format=\"csr\")\n", " \n", " " ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.3" } }, "nbformat": 4, "nbformat_minor": 2 }
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 { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "DATA PRODUCTION\n", "\n", "We define some constants for the whole problem, as well as some functions :\n", "\n", "- wavevector\n", "- potential\n", "- initialisation of E" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "ename": "ModuleNotFoundError", "evalue": "No module named 'CrankNicolson'", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mModuleNotFoundError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mnumpy\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0;32mimport\u001b[0m \u001b[0mCrankNicolson\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 3\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 4\u001b[0m \u001b[0mlambd\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m0.7e-6\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 5\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;31mModuleNotFoundError\u001b[0m: No module named 'CrankNicolson'" ] } ], "source": [ "import numpy as np\n", "import CrankNicolson\n", "\n", "lambd = 0.7e-6\n", "\n", "k = 2 * np.pi / lambd\n", "\n", "w0 = 1e-3\n", "Pin = 1\n", "\n", "diff_coeff = 1j*1/(2 * k)\n", "#diff_coeff = 0\n", "#diff_coeff = 1e-4\n", "\n", "print('diff :', diff_coeff)\n", "\n", "\n", "def potential(r):\n", " try:\n", " return diff_coeff / r\n", " except ZeroDivisionError:\n", " return 0\n", "\n", "\n", "def gaussian(r, r0=0, w0=1, Pin=1):\n", " return np.sqrt(2*Pin/(np.pi*w0**2)) * np.exp(-(r-r0)** 2/(w0**2))\n", "\n", "\n", "def initial_enveloppe(r_pts, w0, Pin):\n", " return np.array([gaussian(r_pts[i], w0=w0, Pin=Pin) for i in range(len(r_pts))])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Instanciation of the CN Class" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "ename": "NameError", "evalue": "name 'CrankNicolson' is not defined", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mcrank\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mCrankNicolson\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 2\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 3\u001b[0m \u001b[0mcrank\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_grid\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mr_max\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m1e-2\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mn_r\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m100\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mz_min\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mz_max\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m10\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mn_z\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m200\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 4\u001b[0m \u001b[0mcrank\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_parameters\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mD\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mdiff_coeff\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mV\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mpotential\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 5\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;31mNameError\u001b[0m: name 'CrankNicolson' is not defined" ] } ], "source": [ "crank = CrankNicolson()\n", "\n", "crank.set_grid(r_max=1e-2, n_r=100, z_min=0, z_max=10, n_z=200)\n", "crank.set_parameters(D=diff_coeff, V=potential)\n", "\n", "crank.solve(initial_enveloppe(crank.r_pts, w0, Pin))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Save the result for later analysis" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "np.savetxt(\"../CN_cylindric_complex_E.dat\", np.abs(crank.E_matrix))\n", "np.savetxt(\"../CN_cylindric_complex_r_pts.dat\", crank.r_pts)\n", "np.savetxt(\"../CN_cylindric_complex_z_pts.dat\", crank.z_pts)" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.3" } }, "nbformat": 4, "nbformat_minor": 2 }
 { "cells": [], "metadata": {}, "nbformat": 4, "nbformat_minor": 2 }
 ... ... @@ -4,7 +4,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ "DATA PRODUCTION\n", "# Data Production\n", "\n", "We define some constants for the whole problem, as well as some functions :\n", "\n", ... ... @@ -15,17 +15,9 @@ }, { "cell_type": "code", "execution_count": 5, "execution_count": null, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "diff : 5.570423008216336e-08j\n" ] } ], "outputs": [], "source": [ "import numpy as np\n", "from CCCN import CrankNicolson\n", ... ... @@ -68,18 +60,9 @@ }, { "cell_type": "code", "execution_count": 6, "execution_count": null, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/home/antoine/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:20: RuntimeWarning: divide by zero encountered in true_divide\n", "/home/antoine/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:20: RuntimeWarning: invalid value encountered in true_divide\n" ] } ], "outputs": [], "source": [ "crank = CrankNicolson()\n", "\n", ... ... @@ -98,7 +81,7 @@ }, { "cell_type": "code", "execution_count": 7, "execution_count": null, "metadata": {}, "outputs": [], "source": [ ... ...
 """ Solution of filamentation with charge density coupled """ import numpy as np import scipy.sparse import scipy.linalg as la import matplotlib.pyplot as plt %matplotlib inline class CrankNicolsonFilamentation: """A class that solves the entire filamentation""" # Cylindrical grid with rectangular time def set_grid(self, r_max, n_r, z_min, z_max, n_z, t_min, t_max, n_t): self.r_max, self.n_r = r_max, n_r self.z_min, self.z_max, self.n_z = z_min, z_max, n_z self.t_min, self.t_max, self.n_t = t_min, t_max, n_t self.r_pts, self.delta_r = np.linspace(0, r_max, n_r, retstep=True, endpoint=False, dtype=complex) self.z_pts, self.delta_z = np.linspace(z_min, z_max, n_z, retstep=True, endpoint=False, dtype=complex) self.t_pts, self.delta_t = np.linspace(t_min, t_max, n_t, retstep=True, endpoint=False, dtype=complex) # Parameters of the scheme # Dr for diffraction, Dt for dispersion def set_parameters(self, Dr, Dt, V, ofi, ava, K, sigma, omega0, tau, f): # V has to be vectorised self.Dr, self.Dt, self.V, self.ofi, self.ava, self.K, self.sigma, self.omega0, self.tau, self.f = \ Dr, Dt, V, ofi, ava, K, sigma, omega0, tau, f # One solving step for r dependency and t dependency def solve(self, E_init): # Coefficient of matrices sig_r = self.Dr * (self.delta_z) / 2. / self.delta_r**2 sig_t = self.Dt * (self.delta_z/2) / 2. / (self.delta_t)**2 # (half t operator) nu = lambda x : -self.V(x) * (self.delta_z) / 4. / self.delta_r # minus sign for V convention # Empty solution matrices (z,r,t) self.E_matrix = np.zeros([self.n_z, self.n_r, self.n_t], dtype=complex) self.rho_matrix = np.zeros([self.n_z, self.n_r, self.n_t]) # Sparse solver Ar = self._fillAr_sp(sig_r, nu, self.n_r) Br = self._fillBr_sp(sig_r, nu, self.n_r) At = self._fillAt_sp(sig_t, self.n_t) # null advection coefficient Bt = self._fillBt_sp(sig_t, self.n_t) # null advection coefficient # Set boundary conditions # Spatial: # Dirichlet at infinity Ar[1,-1] = 1.0 Ar[2,-2] = 0.0 Br[-1,-1] = 0.0 Br[-1,-2] = 0.0 # Neumann at r=0 Ar[0,1] = -2*sig_r Br[0,1] = 2*sig_r # Time # Neumann at t=t_min, t_max for b in [0,1]: At[1,-b] = 1.0 At[2*b,1-3*b] = 0.0 Bt[-b,-b] = 0.0 Bt[-b,1-3*b] = 0.0 # Propagate E = E_init.copy() E1 = E_init.copy() E2 = E_init.copy() rho = np.zeros([self.n_r, self.n_t]) for n in range(self.n_z): # General step initialisation self.E_matrix[n,:,:] = E self.rho_matrix[n,:,:] = rho # First fourth step # t solver for each r with half non linear fE1 = self.f(E) - self.sigma/2*(1 + 1j*self.omega0*self.tau) * rho[:,:] * E[:,:] print('Defocus plasma :', np.amax(np.abs(self.sigma/2*(1 + self.omega0*self.tau) * rho[:,:] * E[:,:])**2)) #print('fE1 :', fE1) #print('max fE1 :', np.amax(fE1)) # Non linear term at origin if n==0: fE_old = fE1.copy() # Half non linear term with half sum for i in range(self.n_r): E1[i,:] = la.solve_banded((1,1), At, Bt.dot(E[i,:]) + \ (self.delta_z/2.) * (1.5 * fE1[i,:] - 0.5 * fE_old[i,:]), \ check_finite=False) # Non linear term update fE_old = fE1.copy() # Middle half step # r solver solver for each t for i in range(self.n_t): E2[:,i] = la.solve_banded((1,1), Ar, Br.dot(E1[:,i]), \ check_finite=False) # Second fourth step # t solver for each r with half non linear fE2 = self.f(E2) - self.sigma/2*(1 + 1j*self.omega0*self.tau) * rho[:,:] * E2[:,:] #print('fE2 :', fE2) #print('max fE2 :', np.amax(fE2)) # Half non linear term with half sum (old already defined even at n=0) for i in range(self.n_r): E[i,:] = la.solve_banded((1,1), At, Bt.dot(E2[i,:]) + \ (self.delta_z/2.) * (1.5 * fE2[i,:] - 0.5 * fE_old[i,:]), \ check_finite=False) # Non linear term update fE_old = fE2.copy() # E has been updated, put into E_matrix at next step # Density coupling (notations article) # Vectorised Calculation for all r # Time integration from null conditions for k in range(self.n_t-1): # Trapeze integral a = np.exp(self.ava*self.delta_t * (np.abs(E[:,k+1])**2 + np.abs(E[:,k])**2)*0.5) # Next t step calculation for next z step rho[:,k+1] = a * (rho[:,k] + (self.delta_t/2) * self.ofi*np.abs(E[:,k])**(2*K)) \ + (self.delta_t/2) * self.ofi*np.abs(E[:,k+1])**(2*K) # rho has been updated, put into rho_matrix at next step # Deep copy of results for export # np.savetxt ? def get_data(self): return self.E_matrix.copy(), self.rho_matrix.copy() # Diagonal packing for banded r def _fillAr_sp(self, sig, nu, n): """Returns a tridiagonal matrix in compact form ab[1+i-j,j]=a[i,j]""" A = np.zeros([3,n], dtype=complex) # A has three diagonals and size n # Superdiagonal A[0,1:] = -(sig - nu(self.r_pts[:-1])) # Diagonal A[1] = 1+2*sig # Subdiagonal A[2,:-1] = -(sig + nu(self.r_pts[1:])) return A # Diagonal packing for banded t def _fillAt_sp(self, sig, n): """Returns a tridiagonal matrix in compact form ab[1+i-j,j]=a[i,j]""" _o = np.ones(n, dtype=complex) A = np.zeros([3,n], dtype=complex) # A has three diagonals and size n # Superdiagonal A[0,1:] = -sig # Diagonal A[1] = 1+2*sig # Subdiagonal A[2,:-1] = -sig return A # Sparse tridiagonal storage for r def _fillBr_sp(self, sig, nu, n): """Returns a tridiagonal sparse matrix in csr-form""" _o = np.ones(n, dtype=complex) supdiag = (sig - nu(self.r_pts[:-1])) diag = (1-2*sig)*_o subdiag = (sig + nu(self.r_pts[1:])) return scipy.sparse.diags([supdiag, diag, subdiag], [1,0,-1], (n,n), format="csr") # Sparse tridiagonal storage for t def _fillBt_sp(self, sig, n): """Returns a tridiagonal sparse matrix in csr-form""" _o = np.ones(n, dtype=complex) supdiag = sig*_o[:-1] diag = (1-2*sig)*_o subdiag = sig*_o[:-1] return scipy.sparse.diags([supdiag, diag, subdiag], [1,0,-1], (n,n), format="csr")
 { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Functions" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "def potential(r):\n", " try:\n", " return diff_coeff / r\n", " except ZeroDivisionError:\n", " return 0\n", "\n", "\n", "def gaussian(r, t, r0=0, w0=1, t0=0, tp=1, Pin=1):\n", " return np.sqrt(2*Pin/(np.pi*w0**2)) * np.exp(-(r-r0)** 2/(w0**2) - (t-t0)** 2/(tp**2))\n", "\n", "\n", "def initial_enveloppe(r_pts, t_pts, w0, tp, Pin):\n", " return np.array([[gaussian(r_pts[i], t_pts[j], w0=w0, tp=tp, Pin=Pin) for j in range(len(t_pts))] \\\n", " for i in range(len(r_pts))])\n", "\n", "#----- Non linearities --------------\n", "\n", "def kerr(E):\n", " return 1j * k * n2 * np.abs(E)**2 * E\n", "\n", "def answer(t):\n", " pass\n", " \n", "def kerr_delay(E):\n", " pass\n", " \n", "\n", "def kerr_MPA(E):\n", " return 1j*k*n2*np.abs(E)**2 * E - beta/2 * np.abs(E)**(2*K-2) * E" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Parameters" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "from FCN import CrankNicolsonFilamentation\n", "\n", "# Beam\n", "c = 3e8\n", "lambd = 0.7e-6\n", "w0 = 1e-3\n", "tp = 85e-15\n", "#tp = 1\n", "Pcr = 1.8e9\n", "#Pin = 5 * Pcr\n", "Pin = 2*Pcr\n", "\n", "# Dispersion\n", "k = 2 * np.pi / lambd\n", "k_2 = 2e-28\n", "\n", "# Kerr Effect\n", "n2 = 5.57e-23\n", "#n2 = 7e-23\n", "\n", "# MPA\n", "K = 7\n", "beta = 6.5e-104\n", "\n", "# Density\n", "omega0 = k * c\n", "ofi = 8.84e-71/omega0\n", "#ofi = 0\n", "ava = 2.9e-6\n", "#ava = 0\n", "sigma = 5.1e-24\n", "tau = 3.5e-13\n", "\n", "diff_coeff = 1j*1/(2 * k)\n", "#diff_coeff = 0\n", "#diff_coeff = 1e-4\n", "disp_coeff = -1j*k_2/2\n", "#disp_coeff = 1j * 1e-25\n", "#disp_coeff = 0\n", "\n", "print('diff :', diff_coeff)\n", "print('disp :', disp_coeff)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Instantiation" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "crank = CrankNicolsonFocus()\n", "\n", "crank.set_grid(r_max=2.8e-3, n_r=100, z_min=0, z_max=10, n_z=100, t_min=-5e-13, t_max=5e-13, n_t=100)\n", "crank.set_parameters(Dr=diff_coeff, Dt=disp_coeff, V=potential, ofi=ofi, ava=ava ,K=K, sigma=sigma, \\\n", " omega0=omega0, tau=tau, f=kerr_MPA)\n", "\n", "crank.solve(initial_enveloppe(crank.r_pts, crank.t_pts, w0, tp, Pin))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Initial conditions" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "mat = initial_enveloppe(crank.r_pts, crank.t_pts, w0, tp, Pin)\n", "\n", "fig, ax = plt.subplots()\n", "mesh = ax.pcolormesh(np.real(crank.t_pts), np.real(crank.r_pts), np.abs(mat)**2)\n", "plt.colorbar(mesh, ax=ax)\n", "#ax.set_ylim(0,1e-2)\n", "fig.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Plots" ] },