# LSA Assignment 4 - Time-dependent problems#

The deadline for submitting this assignment is **Midnight Friday 30 August 2024**.

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Write your code and answers in a Jupyter notebook, then select File -> Download as -> PDF via LaTeX (.pdf).

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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%.