last error

book/fvm/advection_1.ipynb

@@ -414,61 +414,32 @@
     "\n"
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     "## Stability\n",
     "\n",
-    "**Method of Lines** converts partial differential equation (PDE) into a system of ordinary differential equations (ODEs). \n",
+    "This equation can be rewritten to isolate the amplification factor: \n",
     "\n",
     "$$\n",
-    "\\frac{\\phi_{i}^{n+1}-\\phi_{i}^{n}}{\\Delta t}\n",
-    "=  D  \\frac{ \\phi_{i-1}- 2\\phi_i  + \\phi_{i+1} }{\\Delta x^2} \n",
+    "\\phi_i^{n+1} = (1-2*\\Delta t D/\\Delta x^2)\\phi_i^n + \\Delta t D \\frac{\\phi_{i-1}^n + \\phi_{i+1}^n}{\\Delta x^2}\n",
     "$$\n",
     "\n",
-    "**Time Integration**: We can now apply time-stepping methods like Forward Euler or Backward Euler to integrate these ODEs over time.\n",
+    "where the amplification factor is multiplying $\\phi_i^n$ and should be lower than 1 to ensure stability:\n",
     "\n",
-    "   - **Forward Euler (Explicit)**:\n",
-    "\n",
-    "     $$\n",
-    "     \\phi_i^{n+1} =  \\phi_i^n + \\Delta t D \\frac{\\phi_{i-1}^n - 2\\phi_i^n + \\phi_{i+1}^n}{\\Delta x^2}\n",
-    "     $$\n",
-    "\n",
-    "      This can be rewritten as      \n",
-    "      $$\n",
-    "      \\phi_i^{n+1} = (1-2*\\Delta t D/\\Delta x^2)\\phi_i^n + \\Delta t D \\frac{\\phi_{i-1}^n + \\phi_{i+1}^n}{\\Delta x^2}\n",
-    "      $$\n",
+    "$$\n",
+    "-1 <  1-2D \\frac{\\Delta t}{\\Delta x^2} < 1\n",
+    "$$\n",
     "\n",
-    "      Therefore, the amplification factor should be lower than 1:\n",
+    "thus \n",
     "\n",
-    "      $$\n",
-    "      -1 <  1-2D \\frac{\\Delta t}{\\Delta x^2} < 1\n",
-    "      $$\n",
-    "      \n",
-    "      $$\n",
-    "      D \\frac{\\Delta t}{\\Delta x^2} <= 1/2\n",
-    "      $$\n",
+    "$$\n",
+    "D \\frac{\\Delta t}{\\Delta x^2} <= 1/2\n",
+    "$$\n",
     "\n"
    ]
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