PDE I, Spring, 1999
Backward Euler method for the heat equation. This method is unconditionally stable and unconditionally monotone. However it is only first order accurate in time.
Crank-Nicholson method for the heat equation. This is second order in time and unconditionally stable. However, it is only conditionally monotone, as is demonstrated by this jump initial data.
To learn more about this and other facts about PDE, take PDE I, Math 232 in the spring of 1999.