1. Using linear stability analysis, identify any restrictions on the step size for a stable solution.

(a) Consider a lincar first order differential equation:

-10y; y(0) = 10.

(1)

i. Clearly identifying your notation for discrete points, obtain a finite-difference ap-

proximation for the above equation using Forward Euler scheme. Use At as your

time-step for a uniformly spaced grid in time.

ii. Obtain the amplification factor for the above finite difference approximation.

iii. Identify if the above scheme is unconditionally unstable, conditionally stable, or

unconditionally stable. If conditionally stable, obtain any time-step restrictions for

which the scheme will produce stable result. Show all steps.

(b) Consider a linear first order differential equation:

dy

dt

=(-2+2i)y; y(0)=1; i =√1

(2)

Repeat parts (i) (iii) above; i,e. find the amplification factor for the Forward Euler, and

any restrictions on the step sizes for the Forward Euler method.