stability analysis in Unsteady state

1. Explicit Method :

In case of a transient explicit method, stability largely depends on CFL number

CFL number for 2-dimensional heat conduction problem is given as:

k= alpha* (dT/dx² + dT/dy²) ≤ 0.5

in this case dx = dy hence CFL ≤ 0.25, where alpha is thermal diffusivity which is given, hence stability depends on the value of dT and dx. so in the case of a transient explicit scheme, the values of dT and dx should be chosen wisely otherwise the system goes unstable.

the system goes unstable if k ≥ 0.25

and the unstable system is given in the figure below and at the same time step, the implicit method is still stable while the explicit is unstable.

2. Implicit Method:

In case of a transient implicit scheme, the system remains stable if k ≥ 0.25, if you increase the dt value the system still remains stable and the solution is unconditionally stable. the implicit method allows higher time step compare to an explicit method and the solution will still remain stable.

overall to determine which method should be used whether it is an explicit method or implicit method depends largely on the goal of computation. if time accuracy is required then explicit method shows greater accuracy with less computational time.