= BDF2 discrete energy identity
{c}
{title2=$\mathcal E_n$}
For three <vectors> $a,b,c$ in an <inner-product space>, expansion of each squared <norm> proves
$$
2\operatorname{Re}\langle3a-4b+c,a\rangle
=\|a\|^2+\|2a-b\|^2-\|b\|^2-\|2b-c\|^2+\|a-2b+c\|^2.
$$
Therefore the second-order <backward differentiation formula> $3U^{n+2}-4U^{n+1}+U^n=2kLU^{n+2}$ with a <dissipative operator> $L$ decreases the discrete <energy> $\mathcal E_n=\|U^{n+1}\|^2+\|2U^{n+1}-U^n\|^2$. This <quadratic form> is equivalent to the product <norm> of the two time levels with constants independent of $k$ and $L$. It proves uniform <stability> even when amplification <roots of a polynomial> coalesce inside the unit disk, where an eigenvector-separation proof may lose its bound.
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