Question

4. Consider the differential equation dy/dt = −ay with a > 0 an (unspecified) constant.

(a) Write out the Euler step (i.e. yk as a function of yk−1) for the initial value problem y(t0) = y0, with arbitrary step size ∆t.

(b) Find a number u > 0 such that if ∆t > u, then |yk| diverges to +∞ as k goes to +∞. The number u should be expressed as a function of the constant a.

(c) Find another number v > 0 such that if u > ∆t > v, then |yk| converges to zero as k goes to +∞, but yk and yk+1 alternate in signs for each k (even though the true solution does not oscillate). How does v depend on a?

(d) How large can you make the time step ∆t for this system without encountering numerical instabilities or spurious oscillations?

Answer 1