Testing Neumann BCs

Neumann.ipynb

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   "execution_count": 1,
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   "outputs": [],
   "source": [
    "using PDEOperators"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "100-element Array{Float64,1}:\n",
       " -48.7525     \n",
       "   1.08247e-15\n",
       "  -2.15106e-16\n",
       "  -2.70617e-16\n",
       "   1.59595e-16\n",
       "  -3.1225e-16 \n",
       "   9.36751e-17\n",
       "  -5.89806e-17\n",
       "  -5.89806e-17\n",
       "   9.36751e-17\n",
       "  -9.02056e-17\n",
       "  -6.76542e-17\n",
       "   1.17961e-16\n",
       "   ⋮          \n",
       "   3.07046e-16\n",
       "  -6.17562e-16\n",
       "   5.20417e-16\n",
       "  -7.56339e-16\n",
       "   6.73073e-16\n",
       "  -5.75928e-16\n",
       "  -1.59595e-16\n",
       "   0.158912   \n",
       "  -6.76542e-16\n",
       "  -6.76542e-16\n",
       "  -6.76542e-16\n",
       " -52.5        "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = LinearOperator{Float64}(4,4,100,:Neumann,:Neumann;bndry_fn=(2,3)); # 1st derivative values at the boundaries\n",
    "x = collect(0 : 1/(100-1) : 1);\n",
    "u0 = -(x - 0.5).^2 + 1/12;\n",
    "res = A*u0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8-element Array{Float64,1}:\n",
       "  -2.59286 \n",
       "   7.0     \n",
       " -10.5     \n",
       "  11.6667  \n",
       "  -8.75    \n",
       "   4.2     \n",
       "  -1.16667 \n",
       "   0.142857"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "aorder = A.boundary_length - 1;\n",
    "T = Float64;\n",
    "grid_step = one(T);\n",
    "first_order_coeffs = PDEOperators.calculate_weights(1, (0)*grid_step, collect(zero(T) : grid_step : aorder* grid_step))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "126.4083333333333"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(first_order_coeffs.*res[1:8]) # the first derivative at x1, should be equal to 2?"
   ]
  }
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