Stability analysis in the unsteady problem
Stability Analysis:
The Stability of a Numerical Scheme depends on how the error propagates in the equation at different time steps .
If the error magnifies during the course of equation solving then it causes the solution to get unstable .
If the error remains stable during the equation solving then it remains stable.
An important criterion which indicates at the stability of an equation is the Courant Friedrichs Lewy Condition or CFL condition which determines conditions necessary for convergence while solving certain partial differential equations numerically by finite differences methods.
The stability of the 2D Heat Conduction equation is :`C = alpha*(1/(Deltax)^2 + 1/(Deltay)^2)*Deltat`
`alpha`= Thermal Diffusibility(`m^2s^-1`)
`Deltax=Deltay` is Grid points`
`Deltat`= Time step`
where `C<=C_(max)=0.25` for Convergence
Explicit Form :
The stabiliy of Explicit form is dependent on the CFL number which should be less than 0.25.
- For CFL number less than 0.25: dt=30
Number of iteration done are 1000
Time taken for solving equation is 0.018894
cfl number 0.249
- For CFL number more than 0.25: dt=33
Number of iteration done are 1000
Time taken for solving equation is 0.019203
cfl number 0.2740
Discussion:
As the CFL number becomes more than 0.25 we get an unstable solution. With CFL number less than 0.25 we get a stable solution.
Implicit Form:
- Implicit form is stable for large values of dt and the value of C ie CFL number does not affect the stabilty of the solution.
- Implicit Schemes time accuracy decreases with increase in time steps.
- Implicit Schemes are unconditionally stable and it uses entire domain to calculate each time steps therefore not affected by the CFL number.