Commit 2edecd00 authored by Mathieu RASSON's avatar Mathieu RASSON

Half step dispersion term

parents 300858f1 0825fa2f
......@@ -29,7 +29,9 @@
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"import numpy as np\n",
......@@ -168,7 +170,9 @@
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"import numpy as np\n",
......@@ -285,7 +289,7 @@
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......
{
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"cell_type": "markdown",
"metadata": {},
"source": [
"# Crank-Nicholson scheme"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Time dispersion term"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Add time dispersion term as a new diffusion source $D_2$.\n",
"* Add one dimension to tables;\n",
"* Add one parameter to initialisation;\n",
"* Border conditions: null conditions at $t_{min}$ and $t_{max}$;\n",
"* Fill another scheme matrix;\n",
"* Split step into two half steps: fill one dimension for each step inside the $z$ loop, separately."
]
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"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 + f(E)\"\"\"\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, f):\n",
" \n",
" # V has to be vectorised\n",
" self.D, self.V, self.f = D, V, f\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 = -self.V * self.delta_z / 4. / self.delta_r # 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\n",
" for n in range(self.n_z):\n",
" self.E_matrix[n,:] = E\n",
" fE = f(E)\n",
" # Non linear term at origin\n",
" if n==0:\n",
" fE_old = fE\n",
" \n",
" # Non linear term with half sum\n",
" E = la.solve_banded((1,1),A, B.dot(E) + self.delta_z * (1.5 * fE - 0.5 * fE_old),\\\n",
" check_finite=False)\n",
" fE_old = fE\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": "markdown",
"metadata": {},
"source": [
"## Non linear terms"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Compute $\\mathcal{E} \\mapsto f(\\mathcal{E})$ to add non linear terms. Test one after another, with divergence if only Kerr effect."
]
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......@@ -31,7 +31,9 @@
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"metadata": {
"collapsed": true
},
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"import numpy as np\n",
......@@ -300,7 +302,9 @@
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......@@ -321,7 +325,7 @@
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......
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