Mid term project -Stability analysis in the unsteady problem

Implicit method :-

• When a direct computation of the dependent variables can be made in terms of known quantities, the computation is said to be explicit. When the dependent variables are defined by coupled sets of equations, and either a matrix or iterative technique is needed to obtain the solution, the numerical method is said to be implicit.
• In computational fluid dynamics, the governing equations are nonlinear, and the number of unknown variables is typically very large.
• In transient mplicit case the system remains stable if k> 0.25 if you increase the dt value the system stil remains stable and the solution is unconditinally stable the implicit method allow higher time step compareto an explicit method and the solution will remainstable.

Explicit method :-

• In this case a transient method , stability largely depends on CFL number .
• CFL num for 2-dimensional heat conduction problem is given :
• K = alpha *(dT/dx^2+dT/dy^2)<=0.5
• In this case dx = dy hence cfl <= 0.25 where alpha is thermal diffusivity of gold , hence stability depends on the value of dT and dx . so in the case of a transient explicit scheme,n the values of dT and dx should be chosen wisely otherwise the system goes unstable .
• The system goes unstable if k >=0.25.

Conclusion :-

• Therefore we have determinine this method should be used wheather it is an explicit mthod or implicit method depends largely on the goal of computation . it time accuracy is required then explicit method show greater accuracy with less computational time .