First we have implementations for the heat equation in 1D and 2D using the finite difference method and explicit Euler time stepping. The implementations are found in heat-eqn-1d.py and heat_eqn_2d.py
We also use implement the Implicit and Explicit Closest Point Methods from Macdonald and Ruuth's paper https://steveruuth.org/wp-content/uploads/2020/10/icpm.pdf We utilize the method to solve the 2d heat equation, the shifted Poisson equation, and the regular Poisson equation (see files in CPM)