p01.ipynb

{
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   "source": [
    "# p01: Convergence of fourth order finite differences"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us compute the derivative of\n",
    "$$\n",
    "u(x) = \\exp(\\sin(x)), \\qquad x \\in [-\\pi,\\pi]\n",
    "$$\n",
    "using fourth order finite difference scheme\n",
    "$$\n",
    "u'(x_j) \\approx w_j = \\frac{1}{h} \\left( \\frac{1}{12} u_{j-2} - \\frac{2}{3} u_{j-1} + \\frac{2}{3} u_{j+1} - \\frac{1}{12} u_{j+2} \\right)\n",
    "$$\n",
    "using periodic boundary conditions."
   ]
  },
  {
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   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format='svg'\n",
    "from scipy.sparse import coo_matrix\n",
    "from numpy import arange,pi,exp,sin,cos,ones,inf\n",
    "from numpy.linalg import norm\n",
    "from matplotlib.pyplot import figure,loglog,semilogy,text,grid,xlabel,ylabel,title"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
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    "Nvec = 2**arange(3,13)\n",
    "for N in Nvec:\n",
    "    h = 2*pi/N\n",
    "    x = -pi + arange(1,N+1)*h\n",
    "    u = exp(sin(x))\n",
    "    uprime = cos(x)*u\n",
    "    e = ones(N)\n",
    "    e1 = arange(0,N)\n",
    "    e2 = arange(1,N+1); e2[N-1]=0\n",
    "    e3 = arange(2,N+2); e3[N-2]=0; e3[N-1]=1;\n",
    "    D = coo_matrix((2*e/3,(e1,e2)),shape=(N,N)) \\\n",
    "        - coo_matrix((e/12,(e1,e3)),shape=(N,N))\n",
    "    D = (D - D.T)/h\n",
    "    error = norm(D.dot(u)-uprime,inf)\n",
    "    loglog(N,error,'or')\n",
    "    \n",
    "semilogy(Nvec,Nvec**(-4.0),'--')\n",
    "text(105,5e-8,'$N^{-4}$')\n",
    "grid(True); xlabel('N'); ylabel('error')\n",
    "title('Convergence of fourth-order finite difference');"
   ]
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   "source": [
    "## Exercise\n",
    "\n",
    "Find sixth and eigth order finite differences, implement them and make error plot as above.\n",
    "\n",
    "## Exercise\n",
    "\n",
    "Construct the differentiation matrix using `toeplitz` function. You need to pass the first column and first row."
   ]
  }
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