Numerical Analysis class Prove that Eulers method is stable

Numerical Analysis class:

Prove that Euler\'s method is stable with respect to perturbations in the initial data y_0 and the function f. That is, prove that if yn is defined by Euler\'s method and yn is defined by the perturbed equations: y_n + 1 bar = y_n bar + h_f bar + (t_n, y_n bar), y_0 bar is given, then |y_n - y_n bar| lessthanorequalto C_1 |y_0 - y_0 bar| + C_2 ||f- f bar||_L^infinity (I times R). State precisely the hypotheses needed and give explicit formulas for C_1 and C_2.

Solution

There are 2 notions for numerical stability.

1)behaviour of numerical solution for a value of t>0 and h tends to 0.

The numerical approximation doesnot diverge away from true solution if there is a small change in the initial conditions.

2)behaviour of solution when t tends to infinity. and with fixed step size h.

all problems with exponentially decaying solutions are stable. functions with exponetially decaying solutions are of the form f(t,y)=zy(t) with z< or equal to 0.