Added assignment 4 and small bug fixes.

hpc_lecture_notes/2023-assignment_4.md

@@ -0,0 +1,19 @@
+# Assignment 4 - Time-dependent problems
+
+This assignment makes up 30% of the overall marks for the course. The deadline for submitting this assignment is **5pm on 14 December 2023**.
+
+Coursework is to be submitted using the link on Moodle. You should submit a single pdf file containing your code, the output when you run your code, and your answers
+to any text questions included in the assessment. The easiest ways to create this file are:
+
+- Write your code and answers in a Jupyter notebook, then select File -> Download as -> PDF via LaTeX (.pdf).
+- Write your code and answers on Google Colab, then select File -> Print, and print it as a pdf.
+
+Consider a square plate with sides $[−1, 1] × [−1, 1]$. At time t = 0 we are heating the plate up
+such that the temperature is $u = 5$ on one side and $u = 0$ on the other sides. The temperature
+evolves according to $u_t = \Delta u$. At what time $t^*$ does the plate reach $u = 1$ at the center of the plate?
+Implement a finite difference scheme and try with explicit and implicit time-stepping. Numerically investigate the stability of your schemes.
+By increasing the number of discretisation points demonstrate how many correct digits you can achieve. Also,
+plot the convergence of your computed time $t^*$ against the actual time. To 12 digits the wanted
+solution is $t^* = 0.424011387033$.
+
+A GPU implementation of the explicit time-stepping scheme is not necessary but would be expected for a very high mark beyond 80%.

hpc_lecture_notes/_toc.yml

@@ -40,6 +40,7 @@ parts:
   - file: 2023-assignment_1
   - file: 2023-assignment_2
   - file: 2023-assignment_3
+  - file: 2023-assignment_4
 #  - file: 2022-assignment_4
 #- caption: LSA
 #  chapters:

hpc_lecture_notes/simple_time_stepping.ipynb

@@ -209,13 +209,19 @@
     "$$\n",
     "\n",
     "with $A\\in\\mathbb{R}^{n\\times n}$ diagonalizable. For any eigenpair $(\\lambda, \\hat{U})$ of $A$ satisfying $A\\hat{U} = \\lambda\\hat{U}$ the function $U(t) = e^{\\lambda t}\\hat{U}$ is a solution for this problem.\n",
-    "Therefore, for forward Euler to converge we require that\n",
+    "If the eigenvalues are real and negative we require for forward Euler to converge that\n",
     "\n",
     "$$\n",
     "\\Delta t < \\frac{2}{|\\lambda_{max}(A)|},\n",
     "$$\n",
     "\n",
-    "where $\\lambda_{max}$ is the largest eigenvalue by magnitude.\n",
+    "where $\\lambda_{max}$ is the largest eigenvalue by magnitude. Note that if the eigenvalues are complex the condition becomes\n",
+    "\n",
+    "$$\n",
+    "\\Delta t < \\frac{2|\\text{Re}(\\lambda)|}{|\\lambda(A)|},\n",
+    "$$\n",
+    "\n",
+    "where $\\lambda$ is the eigenvalue with largest negative real part by magnitude.\n",
     "\n",
     "As example let us take a look at the problem\n",
     "\n",
@@ -266,7 +272,7 @@
     "\\end{align}\n",
     "$$\n",
     "\n",
-    "We now require that $|(1-\\alpha \\Delta t)^{-1}| < 1$. But for $\\alpha>0$ this is always true. Hence, the backward Euler method is unconditionally stable."
+    "We now require that $|(1-\\alpha \\Delta t)^{-1}| < 1$. But for $\\alpha<0$ this is always true. Hence, the backward Euler method is unconditionally stable."
    ]
   },
   {