I have the following problem:
Show that the BDF-algorithm for a constant stepsize $h$ and $r\in{1,2}$ is convergent for all enough smooth right sides $f$.
I already know, that for $r=1$ the BDF-algorithm equals the implicite Euler-algorithm, which is convergent.
So I only have to show that for $r=2$. Could someone help me here?
Since that probably is some homework, I will not complete the proof/calculation for you, but rather explain some stuff and give hints.
First we should get our definitions straight. There are several methods one usually considers when studying numerics for ODEs. The first two categories are one-step methods and multi-step methods. And BDFs are multi-step methods.
All of these methods have in common, that you can't really speak about "convergence of the method" in general. A method can be as good as possible, but applied to on ODE $u'=f$ it might still not give you the desired result. Convergence depends on both the method and the ODE.
So, the natural thing to do is to get new definitions for a method "being good" and "something else" to then result in "giving the desired result" (=convergence).
1) Consistency
Given a general IVP: $$u'(t) = f(t,u(t)),\ t≥t_0 u(t_0)=t_0$$ on an interval $I=[t_0,t_0+T]$ that is separated into time-steps: $$t_0<t_1<…<t_N=t_0+T.$$
A general multi-step method of stage $R$ is defined as $$\sum_{r=0}^{R}α_{R-r}u_{n-r} = h\sum_{r=0}^Rβ_{R-r}f_{n-r}, \qquad f_n:=f(t_n,u_n).$$ with $α_r, β_r∈ℝ$, for $r=0,…,R$. For standardization $α_R=1$. And w.l.o.g $|α_0| + |β_0|\neq0$ (or it would be a multi-step method of stage $R-1$.)
We define the consistency error $τ_n^h$ as $$τ_n^h:= τ^h(t_n):=\frac{1}{h}\sum_{r=0}^Rα_{R-r}u(t_{n-r}) - \sum_{r=0}^Rβ_{R-r}f(t_{n-r},u(t_{n-r})).$$ With that definition a multi-step method is called consistent if $$\max_{t_n}\|τ_n^h\|→0\qquad (h→0),$$ and the method is consistent of order $p$ if $$\max_{t_n}\|τ_n^h\|=\mathcal{O}(h^p)\qquad (h→0).$$
A result that helps proving consistency is:
A multi-step method is consistent iff $$\sum_{r=0}^Rα_{R-r}=0,\quad \sum_{r=0}^Rrα_{R-r}+\sum_{r=0}^Rβ_{R-r} = 0,$$ and the method is exactly of order $p$ if $$C_0=C_1=…=C_p=0,\quad C_{p+1}\neq 0,$$ with $$C_i=(-1)^i\left\{\frac{1}{i!}\sum_{r=0}^Rr^iα_{R-r}+\frac{1}{(i-1)!}\sum_{r=0}^Rr^{i-1}β_{R-r}\right\}.$$ (Looks horrible, but for your methods of order $R=1,2$, these sums are small.)
2) Stability
Given the first characteristic polynomial of a multi-step method $$ρ(λ)=\sum_{r=0}^Rα_rλ^r,$$ a multi-step method is called null-stable if no root of $ρ$ has modulus larger than $1$ and if all roots with modulus $1$ are unique/single.
3) Convergence
If the IVP fulfils the usual Lipschitz-condition and the multi-step method is null-stable, then $consistency$ (of order $p$) is sufficient for convergence (of order $p$).
This is usually meant, if someone speaks about the convergence of a method - with respect to all IVPs that fulfil the Lipschitz-condition.
edit: I didn't notice "for all enough smooth $f$" in your question. Sorry. So most likely you already know the part about "convergence = consistency + stability". So just go ahead and use what I wrote above about consistency and stability on the BDF formula for $R=2$.
