course | course_year | question_number | tags | title | year | |||
---|---|---|---|---|---|---|---|---|
Numerical Analysis |
IB |
60 |
|
Paper 1, Section II, D |
2016 |
(a) Consider a method for numerically solving an ordinary differential equation (ODE) for an initial value problem,
(b) A general multistep method for the numerical solution of an ODE is
$$\sum_{l=0}^{s} \rho_{l} \mathbf{y}{n+l}=h \sum{l=0}^{s} \sigma_{l} \mathbf{f}\left(t_{n+l}, \mathbf{y}_{n+l}\right), \quad n=0,1, \ldots$$
where
(c) State the Dahlquist equivalence theorem regarding the convergence of a multistep method.
(d) Consider the multistep method,
$$\mathbf{y}{n+2}+\theta \mathbf{y}{n+1}+a \mathbf{y}{n}=h\left[\sigma{0} \mathbf{f}\left(t_{n}, \mathbf{y}{n}\right)+\sigma{1} \mathbf{f}\left(t_{n+1}, \mathbf{y}{n+1}\right)+\sigma{2} \mathbf{f}\left(t_{n+2}, \mathbf{y}_{n+2}\right)\right]$$
Determine the values of