+ +
+

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%.

+
+ + + + +