Industrial Mathematics - Alexiades
Homework on Forward and Backward Euler schemes
To get a sense about how numerical schemes for ODEs work,
apply the Forward Euler and the Backward Euler schemes,
BY HAND, to approximate y(2), using Δt=1/2, for:
1. dy/dt = y − 2t , y(0) = 1
2. dy/dt = ty , y(0) = 1
Note that Backward Euler can be done by hand for these,
only because f(t,y) is linear in y, so can easily solve the equation for
Yn+1.
But this would NOT be the case for,
e.g., y′ = t y3ey (try one step!).
In general, in an IMPLICIT scheme, an equation must be solved numerically at each time step
(using a root finder!).
This is just for gaining insight... NOTHING TO TURN IN.
Forward Euler scheme:
Y0 = y0,
Yn+1 = Yn + Δt F(n , Yn), n=0,1,2,...
Backward Euler scheme:
Y0 = y0,
Yn+1 = Yn + Δt F(n+1 , Yn+1), n=0,1,2,...