Heat-wave lets you solve heat and wave sheets with three methods: forward-difference, backward-difference, and central-difference. With backward-difference you can take choose very big time steps. You loose accuracy, but you can quickly get an approximate solution. However, with forward-difference choosing a large time step results in an unstable and non-converging solution.
All of the illustrations on this page were generated by solving the heat equation with rates greater than one half. 1D, rate 1/2 is stable but oscillates. Equation. Greater than 0.5 in unstable. 1/2 overshoots. 1/3 works fine. 1/4 looks like an attractive distribution. 2D case max rate is 0.25. Even backward difference blows up with negative values.